Open Access

An anisotropic quasilinear problem with perturbations

Boundary Value Problems20132013:147

DOI: 10.1186/1687-2770-2013-147

Received: 7 December 2012

Accepted: 30 April 2013

Published: 18 June 2013

Abstract

This work focuses on proving the existence and uniqueness of strong solutions of perturbed anisotropic total variation flow with the Neumann boundary condition when the initial data is an L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif function.

MSC:35K65, 35K55.

Keywords

anisotropic total variation flow semigroups strong solution

1 Introduction

Problems of general anisotropic total variation flow arise in a number of areas of science. The parabolic equations represent what Giga et al. called a very singular diffusivity (see [1]) and are a natural generalization of the total variation flow in the presence of an anisotropy. In the isotropic case, the equation u t = div ( ξ f ( D u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq2_HTML.gif becomes u t = div ( D u | D u | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq3_HTML.gif when the Lagrangian f : R N [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq4_HTML.gif is given by f ( ξ ) = ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq5_HTML.gif, where ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq6_HTML.gif is the usual l 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq7_HTML.gif-norm; i.e., ξ : = ( i = 1 N ξ i 2 ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq8_HTML.gif. Let us recall that this PDE appears when one uses the steepest decent method to minimize the total variation. This method was introduced by Rudin and Osher (see [2, 3]) in the context of image denoising and reconstruction. In the last years, its applications have been studied by many authors (see [47]).

Let Ω be an open bounded subset in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq9_HTML.gif, N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq10_HTML.gif, with boundary Ω of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq11_HTML.gif. In this paper, we are interested in the problem
{ u t = div ( ξ f ( x , D u ) ) h ( x , u ) in  [ 0 , T ] × Ω , u η = 0 on  [ 0 , T ] × Ω , u ( 0 ) = u 0 in  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ1_HTML.gif
(1.1)

where u 0 L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq12_HTML.gif, f ( x , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq13_HTML.gif is a 1-homogeneous convex function with linear growth as ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq14_HTML.gif, η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq15_HTML.gif is the Neumann boundary operator associated to ξ f ( x , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq16_HTML.gif, i.e., u η : = ξ f ( x , ξ ) ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq17_HTML.gif with ν the unit outward normal on Ω, and the function h ( x , u ) : Ω × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq18_HTML.gif satisfies the following assumptions, which we shall refer to collectively as (M):

(M1) For almost all x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq19_HTML.gif, r h ( x , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq20_HTML.gif is continuous nondecreasing, and h ( x , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq21_HTML.gif;

(M2) For every r R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq22_HTML.gif, x h ( x , r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq23_HTML.gif is in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif.

As argued in [8], the choice of Neumann boundary conditions is a natural choice in image processing. It corresponds to the reflection of the picture across the boundary and has the advantage of not imposing any value on boundary and not creating edges on it. For instance, in [9], Andreu, Caselles and Mazón considered the elliptic problem u div a ( u , D u ) = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq24_HTML.gif with Neumann boundary conditions. In [7], Andreu et al. obtained the existence and uniqueness of entropy solutions of quasilinear parabolic equation with the Neumann boundary, i.e.,
{ u t = div a ( u , D u ) in  ( 0 , T ) × Ω , u η = 0 on  ( 0 , T ) × Ω , u ( 0 ) = u 0 in  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ2_HTML.gif
(1.2)

where u ( 0 ) = u 0 L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq25_HTML.gif, a ( z , ξ ) = ξ f ( z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq26_HTML.gif, and f C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq27_HTML.gif satisfies some additional assumptions. Our problem is closely related to motion under anisotropic mean curvature flow (see [10]) when h ( x , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq28_HTML.gif. If we take the f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq29_HTML.gif-distance to give a set E as an initial condition ( f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq29_HTML.gif being the polar function of f), then each sublevel set of the anisotropic mean curvature motion behaves instantaneously as the solution of Cauchy problem (1.1) where Ω = R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq30_HTML.gif. Recently Moll [11] proved the existence and uniqueness of the solutions of Dirichlet problem (1.1) with h ( x , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq28_HTML.gif. As we all know, it is possible that the solution of (1.1) will blow up with perturbations. Therefore, in this paper, we extend the problem introduced in Moll [11] and obtain the existence and uniqueness of strong solutions of (1.1) when perturbation term satisfies assumption (M).

This paper is organized as follows. In Section 2 we recall some notions and basic facts. In Section 3 we define the notion of a strong solution for the Neumann problem of (1.1), and give the basic results in this paper. In Section 4 we prove the existence and uniqueness of solutions of an auxiliary equation, i.e.,
{ u t = div ( ξ f ( x , D u ) ) in  [ 0 , T ] × Ω , u η = 0 on  [ 0 , T ] × Ω , u ( 0 ) = u 0 in  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ3_HTML.gif
(1.3)

and for some h ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq31_HTML.gif we obtain the existence and uniqueness of a strong solution of problem (1.1).

2 Preliminaries

To make precise our notions, let us recall some preliminary facts.

Given u BV ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq32_HTML.gif, Du decomposes into absolutely continuous and singular parts D u = u L N Ω + D s u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq33_HTML.gif, where u denotes the Radon-Nikodým derivative with respect to the Lebesgue measure and D s u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq34_HTML.gif is its singular part. There is also the polar decomposition D s u = D s u | D s u | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq35_HTML.gif, where | D s u | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq36_HTML.gif is the total variation measure of D s u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq34_HTML.gif. For further information concerning functions of bounded variation, we refer to [12].

By L w 1 ( [ 0 , T ] ; BV ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq37_HTML.gif we denote the space of weakly measurable functions w : [ 0 , T ] BV ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq38_HTML.gif (i.e., t [ 0 , T ] w ( t ) , ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq39_HTML.gif is measurable for every ϕ BV ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq40_HTML.gif) such that 0 T w ( t ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq41_HTML.gif. Observe that since BV ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq42_HTML.gif has separable predual, it follows easily that the map t [ 0 , T ] w ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq43_HTML.gif is measurable.

We shall need several results from [13] in order to give sense to the integrals of bounded vector fields with divergence in L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq44_HTML.gif. Let p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq45_HTML.gif and p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq46_HTML.gif be such that 1 p + 1 p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq47_HTML.gif. Following [13], let
X p ( Ω ) = { z L ( Ω , R N ) : div ( z ) L p ( Ω ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equa_HTML.gif
If z X p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq48_HTML.gif and u BV ( Ω ) L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq49_HTML.gif, the functional ( z , D u ) : C 0 ( Ω ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq50_HTML.gif is defined by the formula
( z , D u ) , φ = Ω u φ div ( z ) d x Ω u z φ d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equb_HTML.gif

Then ( z , D u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq51_HTML.gif is a Radon measure in Ω, Ω ( z , D w ) = Ω z w d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq52_HTML.gif for all w W 1 , 1 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq53_HTML.gif, and ( z , D w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq54_HTML.gif is absolutely continuous with respect to D w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq55_HTML.gif with the Radon-Nikodým derivative θ ( z , D w , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq56_HTML.gif which is a D w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq55_HTML.gif measurable function from Ω to such that B ( z , D w ) = B θ ( z , D w , x ) D w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq57_HTML.gif for any Borel set B Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq58_HTML.gif. We also have that θ ( z , D w , ) L ( Ω , D w ) z L ( Ω , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq59_HTML.gif.

In [13], a weak trace on Ω of the normal component of z X p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq48_HTML.gif is defined. Concretely, it is proved that there exists a linear operator γ : X p ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq60_HTML.gif such that γ ( z ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq61_HTML.gif and γ ( z ) ( x ) = z ( x ) ν ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq62_HTML.gif for all x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq63_HTML.gif if z C 1 ( Ω ¯ , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq64_HTML.gif.

Next, let us introduce the concept of generalized total variation of a BV function with respect to a Finsler metric [14]. Let f : Ω ¯ × R N [ 0 , ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq65_HTML.gif be a Borel function not identically +∞. The function f will be called convex if for any x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq66_HTML.gif, the function f ( x , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq67_HTML.gif is convex on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq9_HTML.gif. We shall say that f is lower semicontinuous (in short l.s.c.) if f ( x , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq67_HTML.gif is lower semicontinuous for any x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq66_HTML.gif. The function will be called positively homogeneous of degree 1 (in short 1-homogeneous) if it satisfies the following property:
f ( x , t ξ ) = | t | f ( x , ξ ) , x Ω , ξ R N , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ4_HTML.gif
(2.1)
f is a sublinear growth if there exists a positive constant 0 < C 0 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq68_HTML.gif such that
0 f ( x , ξ ) C 0 ξ , x Ω , ξ R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ5_HTML.gif
(2.2)
Let us recall that f : Ω ¯ × R N [ 0 , ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq69_HTML.gif is a Finsler metric if it is a Borel function and it satisfies (2.1) and (2.2). If f satisfies (2.1), then the dual function f : Ω × R N [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq70_HTML.gif is defined by f ( x , ξ ) = sup { ( ξ , ξ ) : ξ R N , f ( x , ξ ) 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq71_HTML.gif. It is easy to verify that f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq72_HTML.gif is convex, l.s.c. and satisfies (2.1). Then, if we adopt the following conventions: for any a [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq73_HTML.gif, we set a + = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq74_HTML.gif; a 0 = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq75_HTML.gif if a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq76_HTML.gif and 0 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq77_HTML.gif, we get
f ( x , ξ ) = sup { ( ξ , ξ ) f ( x , ξ ) : ξ R N } , x Ω , ξ R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equc_HTML.gif
We say that f is coercive if there exists a positive constant 0 < C 1 < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq78_HTML.gif such that
f ( x , ξ ) C 1 ξ , x Ω , ξ R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ6_HTML.gif
(2.3)

It is easy to see that f is convex and has a sublinear growth, then f ( x , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq67_HTML.gif is continuous for any x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq19_HTML.gif.

We introduce the classes of vector fields
X c p ( Ω ) : = { z X p ( Ω ) : supp ( z )  is compact in  Ω } , H f ( Ω ) : = { z X c p ( Ω ) : f ( x , z ( x ) ) 1  for a.e.  x Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equd_HTML.gif
and
M f p ( Ω ) : = { z X p ( Ω ) : f ( x , z ( x ) ) 1  for a.e.  x Ω } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Eque_HTML.gif
Let u BV ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq32_HTML.gif, the generalized total variation of u with respect to f in Ω is defined by
Ω | D u | f : = sup { Ω ( z , D u ) : z H f ( Ω ) } = sup { Ω u div ( z ) d x : z H f ( Ω ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ7_HTML.gif
(2.4)

As a direct consequence of the definition, we have that the generalized total variation of u with respect to f in Ω is L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq79_HTML.gif-lower semicontinuous in Ω.

Now, we introduce the relaxed functional, which plays a basic role in proving the existence and uniqueness of the problem.

We define the functional G [ f ] : L 1 ( Ω ) [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq80_HTML.gif by
G [ f ] ( u ) : = { Ω f ( x , u ( x ) ) d x if  u W 1 , 1 ( Ω ) , + , otherwise. https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ8_HTML.gif
(2.5)
We denote by G [ f ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq81_HTML.gif the relaxed functional of G [ f ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq82_HTML.gif; i.e.,
G [ f ] ( u ) : = inf { u n } { lim inf n G [ f ] ( u n ) : u n W 1 , 1 ( Ω ) , u n u L 1 ( Ω ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equf_HTML.gif
In [14], Amer and Belletini obtained the following result:
Ω | D u | f = G [ f ] ( u ) = G [ f ] ( u ) , u BV ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ9_HTML.gif
(2.6)
Moreover, in [11], Moll proved the representation result:
Ω | D u | f = Ω [ R ( f ) ] ( x , ν u ) | D u | , u BV ( Ω ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ10_HTML.gif
(2.7)
where [ R ( f ) ] : Ω × R N [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq83_HTML.gif is a representative of the equivalence class of homogeneous integrands h D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq84_HTML.gif associated to sets D K = M f p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq85_HTML.gif which are countable and sequentially weakly-dense in M f p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq86_HTML.gif, and
Ω [ R ( f ) ] ( x , ν u ) | D u | = Ω j ( x , u ( x ) ) d x + Ω [ R ( f ) ] ( x , ν u ) | D s u | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ11_HTML.gif
(2.8)
for all u BV ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq32_HTML.gif, where j ( x , ξ ) = sup { z ( x ) ξ : z M f p ( Ω ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq87_HTML.gif. The following useful inequality holds:
j ( x , ξ ) sup { ξ ξ : ξ R N , f ( x , ξ ) 1 } = f ( x , ξ ) , ( x , ξ ) Ω × R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ12_HTML.gif
(2.9)
The equality holds if and only if the functional G [ f ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq82_HTML.gif defined by (2.5) is L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq88_HTML.gif-lower semicontinuous on W 1 , 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq89_HTML.gif. By the inequality (2.9), we have the measure | D u | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq90_HTML.gif as follows:
| D u | f ( B ) = Ω B | D u | f = Ω B j ( x , u ( x ) ) d x + Ω B [ R ( f ) ] ( x , ν u ) | D s u | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ13_HTML.gif
(2.10)

for every Borel set B R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq91_HTML.gif.

In this paper, we assume that f : Ω × R N [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq92_HTML.gif is a convex homogeneous integrand, i.e., for some constants C 0 , C 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq93_HTML.gif
C 1 ξ f ( x , ξ ) C 0 ξ , ( x , ξ ) Ω × R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ14_HTML.gif
(2.11)
Let us define the functional F ( u ) : L 1 ( Ω ) [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq94_HTML.gif by the formula
F ( u ) : = { Ω f ( x , u ( x ) ) d x , u W 1 , 1 ( Ω )  and  u η = 0  on  Ω , + , otherwise. https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ15_HTML.gif
(2.12)
By u η = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq95_HTML.gif on Ω, and Theorem 4 in [11], it is easy to obtain that the functional F : BV ( Ω ) [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq96_HTML.gif is the relaxed functional of F defined by
F ( u ) : = Ω | D u | f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ16_HTML.gif
(2.13)

3 Strong solutions and main results

In this section we give the main concepts and results of Neumann problems (1.3) and (1.1).

Definition 3.1 A function u C ( [ 0 , T ] ; L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq97_HTML.gif is a strong solution of (1.3) if u W loc 1 , 2 ( [ 0 , T ] ; L 2 ( Ω ) ) L w 1 ( [ 0 , T ] ; BV ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq98_HTML.gif, there exists z L ( [ 0 , T ] × Ω ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq99_HTML.gif with f ( x , z ( t , x ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq100_HTML.gif a.e. in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq9_HTML.gif and a.e. t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq101_HTML.gif, such that
u t = div ( z ) in  D ( [ 0 , T ] × Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ17_HTML.gif
(3.1)
and a.e. t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq101_HTML.gif it holds
Ω ( z ( t ) , D u ( t ) ) = Ω | D u ( t ) | f , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ18_HTML.gif
(3.2)
[ z ( t ) , ν ] = 0 , H N 1 -a.e. on  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ19_HTML.gif
(3.3)

Next we give the main definition in this paper that is the strong solution of problem (1.1).

Definition 3.2 A function u C ( [ 0 , T ] ; L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq97_HTML.gif is a strong solution of (1.1) if u W loc 1 , 2 ( [ 0 , T ] ; L 2 ( Ω ) ) L w 1 ( [ 0 , T ] ; BV ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq102_HTML.gif, there exists z L ( [ 0 , T ] × Ω ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq103_HTML.gif with f ( x , z ( t , x ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq100_HTML.gif a.e. in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq9_HTML.gif and u t = div ( z ) h ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq104_HTML.gif in D ( [ 0 , T ] × Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq105_HTML.gif a.e. t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq101_HTML.gif such that
Ω ( u ( t ) w ) u t Ω ( z ( t ) , D w ) Ω | D u ( t ) | f + Ω ( w u ( t ) ) h ( x , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ20_HTML.gif
(3.4)

for every w W 1 , 1 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq106_HTML.gif and a.e. on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq107_HTML.gif.

The main results of this paper are the following.

Theorem 3.3 Let u 0 L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq12_HTML.gif. Assume that f satisfies (2.11), then there exists a unique strong solution of (1.3) in [ 0 , T ] × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq108_HTML.gif for every T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq109_HTML.gif such that u ( 0 ) = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq110_HTML.gif. Moreover, if u ( t ) ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq111_HTML.gif, u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq112_HTML.gif are the strong solutions of (1.3) corresponding to initial data u 0 ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq113_HTML.gif, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq114_HTML.gif, respectively, then
( u ( t ) ˜ u ( t ) ) + 2 ( u 0 ˜ u 0 ) + 2 for any t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ21_HTML.gif
(3.5)
Theorem 3.4 Let u 0 L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq12_HTML.gif. Assume that f satisfies (2.11) and h ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq31_HTML.gif satisfies (M), then there exists a unique strong solution of (1.1) in [ 0 , T ] × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq108_HTML.gif for every T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq109_HTML.gif such that u ( 0 ) = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq110_HTML.gif. Moreover, if u ˆ ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq115_HTML.gif, u ˆ ( t ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq116_HTML.gif are the strong solutions of (1.1) corresponding to initial data u ˆ 01 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq117_HTML.gif, u ˆ 02 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq118_HTML.gif, respectively, then
( u ˆ ( t ) 1 u ˆ ( t ) 2 ) + 2 ( u ˆ 01 u ˆ 02 ) + 2 for any t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ22_HTML.gif
(3.6)

4 Proof of the main results

In this section we prove Theorem 3.3 by using the techniques of completely accretive operators [15] and Crandall-Liggett’s semigroup generation theorem [16].

Let us recall the notion of completely accretive operators introduced in [15]. Let M ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq119_HTML.gif be the space of measurable functions in Ω. Given u , v M ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq120_HTML.gif, we shall write that u v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq121_HTML.gif if and only if Ω j ( u ) d x Ω j ( v ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq122_HTML.gif for all j J 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq123_HTML.gif, where J 0 = { j : R [ 0 , ] , convex, l.s.c.,  j ( 0 ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq124_HTML.gif. Let A be an operator (possibly multivalued) in M ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq119_HTML.gif, i.e., A M ( Ω ) × M ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq125_HTML.gif. We shall say that A is completely accretive if
u u ˜ u u ˜ + λ ( v v ˜ ) for all  λ > 0  and all  ( u , v ) , ( u ˜ , v ˜ ) A . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equg_HTML.gif
Let P 0 = { p C ( R ) : 0 p 1 , supp ( p )  be compact and  0 supp ( p ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq126_HTML.gif. If A L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq127_HTML.gif, then A is completely accretive if and only if Ω p ( u u ˜ ) ( v v ˜ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq128_HTML.gif for any p P 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq129_HTML.gif, ( u , v ) , ( u ˜ , v ˜ ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq130_HTML.gif. A completely accretive operator in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif is said to be m-completely accretive if R ( I + λ A ) = L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq131_HTML.gif for any λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq132_HTML.gif. In that case, by Crandall-Liggett’s theorem, A generates a contraction semigroup denoted by { S ( t ) } t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq133_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif, which is given by the exponential formula
S ( t ) u 0 = e t A u 0 = lim n ( I + t n A ) n u 0 for any  u 0 L 2 ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equh_HTML.gif
Let us write u ( t ) = e t A u 0 = S ( t ) u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq134_HTML.gif, then u C ( [ 0 , T ] , L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq135_HTML.gif for any T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq136_HTML.gif, and it is a mild solution (a solution in the sense of semigroups [15]) of
d u d t + A u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ23_HTML.gif
(4.1)

such that u ( 0 ) = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq137_HTML.gif.

We shall use a stronger notion of the solution of (4.1). We say that v C ( [ 0 , T ] , L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq138_HTML.gif is a strong solution of (4.1) on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq107_HTML.gif if v W loc 1 , 2 ( ( 0 , T ) , L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq139_HTML.gif and v ( t ) + A v ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq140_HTML.gif for almost all t ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq141_HTML.gif. If u 0 D ( A ) = { u ¯ L 2 ( Ω ) : ( u ¯ , v ¯ ) A  for some  v ¯ L 2 ( Ω ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq142_HTML.gif (the domain of A) and A is m-completely accretive, then u W loc 1 , 2 ( ( 0 , T ) , L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq143_HTML.gif and u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq112_HTML.gif is a strong solution of (4.1) on ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq144_HTML.gif for all T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq145_HTML.gif.

To obtain the solution of problem (1.1), we need the result of problem (1.3). Thus, at first, we will prove the existence and uniqueness of a strong solution of problem (1.3). Let us introduce the following operator A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif associated to problem (1.3).

( u , v ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq147_HTML.gif if and only if u BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq148_HTML.gif, v L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq149_HTML.gif and there exists z M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq150_HTML.gif, v = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq151_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq152_HTML.gif such that
Ω ( w u ) v d x Ω z w d x Ω | D u | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ24_HTML.gif
(4.2)

for all w W 1 , 1 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq153_HTML.gif.

Proposition 4.1 The operator A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif is m-completely accretive with dense domain. For any u 0 L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq154_HTML.gif, the semigroup solution u ( t ) = e t A u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq155_HTML.gif is a mild solution of
{ d u d t + A u 0 , u ( 0 ) = u 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ25_HTML.gif
(4.3)

To prove Proposition 4.1, we need to prove the operator A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif has some characterization, satisfies the range condition and has dense domain in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif.

By the results of Section 2, the relaxed functional is convex and lower semicontinuous. Therefore, the subdifferential F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq156_HTML.gif of is a maximal monotone operator in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif, and consequently, if { T ( t ) } t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq157_HTML.gif is the semigroup solution in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif generated by F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq158_HTML.gif, u ( t ) = T ( t ) u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq159_HTML.gif is a strong solution of the problem (see [15])
{ d u d t + F u ( t ) 0 , t [ 0 , + ] , u ( 0 ) = u 0 , u 0 L 2 ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ26_HTML.gif
(4.4)

Recall that the operator F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq156_HTML.gif is defined by ( u , v ) F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq160_HTML.gif if and only if u , v L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq161_HTML.gif and F ( w ) F ( u ) Ω v ( w u ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq162_HTML.gif, w L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq163_HTML.gif.

To prove the existence and uniqueness of a strong solution of problem (1.3), we also need the next proposition.

Proposition 4.2 The operator F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq156_HTML.gif has dense domain in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif and F = A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq164_HTML.gif.

The following lemmas will be used to prove Proposition 4.1 and Proposition 4.2.

Lemma 4.3 We have the following characterization of the operator A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif, ( u , v ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq147_HTML.gif if and only if u BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq148_HTML.gif, v L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq149_HTML.gif and there exists z M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq150_HTML.gif, v = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq151_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq152_HTML.gif such that
Ω ( w u ) v d x Ω ( z , D w ) Ω | D u | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ27_HTML.gif
(4.5)
for all w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq165_HTML.gif. Moreover, we have that
  1. (i)

    Ω ( z , D u ) = Ω | D u | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq166_HTML.gif,

     
  2. (ii)

    Ω v u = Ω | D u | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq167_HTML.gif,

     
  3. (iii)

    Ω w v = Ω ( z , D w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq168_HTML.gif for all w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq169_HTML.gif.

     

Proof We denote the operator by defined in the statement of the lemma. Since Ω z w = Ω ( z , D w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq170_HTML.gif when w W 1 , 1 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq171_HTML.gif, we have B A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq172_HTML.gif. Let u BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq148_HTML.gif, v L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq149_HTML.gif and there exists z M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq173_HTML.gif, v = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq151_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq174_HTML.gif and (4.2). Let w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq169_HTML.gif, applying results from [13], we have that there exists a sequence w n W 1 , 1 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq175_HTML.gif such that w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq176_HTML.gif in L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq79_HTML.gif, Ω | w n | Ω | D w | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq177_HTML.gif and Ω z w n = Ω ( z , D w n ) Ω ( z , D w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq178_HTML.gif. Using w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq179_HTML.gif as a test function in (4.2) and letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq180_HTML.gif, we obtain (4.5), then we conclude that A B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq181_HTML.gif, therefore A = B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq182_HTML.gif.

Letting w = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq183_HTML.gif in (4.5), we get
Ω ( z , D u ) Ω | D u | f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equi_HTML.gif

Moreover, using Ω | D u | f : = sup { Ω ( z , D u ) : z H f ( Ω ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq184_HTML.gif, we obtain (i).

We take w = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq185_HTML.gif in (4.5) to obtain Ω u v d x Ω | D u | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq186_HTML.gif. Using w = 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq187_HTML.gif in (4.5) and (i), we get
Ω u v d x 2 Ω ( z , D u ) Ω | D u | f = Ω | D u | f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equj_HTML.gif

Thus, (ii) holds.

Using (ii) in (4.5) we have
Ω w v d x Ω ( z , D w ) , w BV ( Ω ) L 2 ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equk_HTML.gif

Since the same inequality holds for w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq188_HTML.gif, (iii) is obtained. □

We consider the following possibly multi-valued functions: A ( x , ξ ) : = ξ f ( x , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq189_HTML.gif and B ( x , ξ ) : = ξ j ( x , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq190_HTML.gif. By the convexity of f, it follows that A is a monotone function satisfying
C 0 δ C 1 , δ A ( x , ξ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ28_HTML.gif
(4.6)
For each x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq66_HTML.gif, we consider the Moreau-Yosida approximation to f ( x ) : R [ 0 , + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq191_HTML.gif defined by
f λ ( x ) ( ξ ) : = min ζ R n { 1 2 λ ζ ξ 2 + f ( x ) ( ζ ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ29_HTML.gif
(4.7)
and the Yosida approximation of the multi-valued function A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq192_HTML.gif is defined as
A λ ( x ) ( ξ ) = I ( I + λ A ( x ) ) 1 λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equl_HTML.gif
We have that ξ f λ ( x ) ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq193_HTML.gif is a convex Fréchet differentiable function (see [17]) such that f λ ( x ) f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq194_HTML.gif pointwise and a.e. in Ω when λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq195_HTML.gif and A λ ( x ) ( ξ ) = f λ ( x ) ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq196_HTML.gif. Moreover, when ζ = J λ ( x ) ( ξ ) = ( I + λ A ( x ) ) 1 ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq197_HTML.gif, we get the minimum in (4.6). In [11], Moll gave the following estimate:
C 1 ξ f ( x , ξ ) f λ ( x ) ( ξ ) { C 0 2 ξ if  ξ C 0 λ , 1 2 λ ξ 2 if  ξ < C 0 λ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ30_HTML.gif
(4.8)
A λ ( x ) ( ξ ) ξ f λ ( x ) ( ξ ) and A λ ( x ) ( ξ ) ξ C 1 ξ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ31_HTML.gif
(4.9)
We consider the operator A n ( x , ξ ) = A 1 n ( x , ξ ) + 1 n ξ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq198_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif to prove Proposition 4.1. Let W p 1 , 2 ( Ω ) : = { w W 1 , 2 ( Ω ) : w η = 0  on  Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq199_HTML.gif, we define ( u , v ) A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq200_HTML.gif if and only if u W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq201_HTML.gif, v L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq202_HTML.gif and
Ω ( w u ) v d x Ω A n ( x ) ( u ) ( w u ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ32_HTML.gif
(4.10)

for all w W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq203_HTML.gif.

The operator A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq204_HTML.gif satisfies the classical Leray-Lions assumption [18]. Hence, for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq205_HTML.gif, the operator A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq204_HTML.gif satisfies L ( Ω ) R ( I + A n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq206_HTML.gif.

Moreover, we need the following characterization of the operator A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq204_HTML.gif.

Lemma 4.4 For every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq205_HTML.gif, the operator A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq204_HTML.gif is completely accretive in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif.

Proof Let p P 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq207_HTML.gif and ( u , v ) , ( u ˜ , v ˜ ) A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq208_HTML.gif. Since ( u , v ) A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq200_HTML.gif, taking w = u p ( u u ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq209_HTML.gif as a test function in (4.10), we get
Ω p ( u u ˜ ) v d x Ω A n ( x ) ( u ) p ( u u ˜ ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ33_HTML.gif
(4.11)
Similarly, ( u ˜ , v ˜ ) A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq210_HTML.gif, we take w = u ˜ + p ( u u ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq211_HTML.gif as a test function in (4.10) and obtain
Ω p ( u u ˜ ) v ˜ d x Ω A n ( x ) ( u ˜ ) p ( u u ˜ ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ34_HTML.gif
(4.12)
Using (4.11) + (4.12), we may write that
Ω p ( u u ˜ ) ( v v ˜ ) d x Ω A n ( x ) ( u u ˜ ) p ( u u ˜ ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equm_HTML.gif
According to (4.6) and u W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq212_HTML.gif, we obtain that q : = A n ( x ) ( u ) X p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq213_HTML.gif. Moreover, by Lemma 4.3 and Theorem 2 in [4], we have that
Ω p ( u u ˜ ) ( v v ˜ ) d x Ω ( q q ˜ ) p ( u u ˜ ) d x = Ω ( ( q q ˜ ) , D p ( u u ˜ ) ) = Ω θ ( ( q q ˜ ) , D p ( u u ˜ ) , x ) | D p ( u u ˜ ) | 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equn_HTML.gif

It follows that the operator A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq204_HTML.gif is completely accretive in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif. □

Lemma 4.5 The operator A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif satisfies L ( Ω ) R ( I + A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq214_HTML.gif, and D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq215_HTML.gif is dense in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif.

Proof We divide the proof into two steps.

Step 1. We first prove L ( Ω ) R ( I + A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq216_HTML.gif. Let v L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq217_HTML.gif, we shall find u BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq218_HTML.gif such that ( u , v u ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq219_HTML.gif, i.e., there exists z X 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq220_HTML.gif such that f ( x , z ( x ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq221_HTML.gif, L N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq222_HTML.gif-a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq223_HTML.gif, v u = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq224_HTML.gif and
Ω ( w u ) ( v u ) d x Ω z w d x Ω | D u | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ35_HTML.gif
(4.13)

for all w W 1 , 1 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq153_HTML.gif.

Using (4.10) and L ( Ω ) R ( I + A n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq206_HTML.gif, we have that for every n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq205_HTML.gif there is u n W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq225_HTML.gif such that ( u n v u n ) A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq226_HTML.gif and
Ω ( w u n ) ( v u n ) d x Ω A n ( x ) ( u n ) ( w u n ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ36_HTML.gif
(4.14)
for all w W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq227_HTML.gif. Since A n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq204_HTML.gif is completely accretive, it is obtained that
u n = ( I + A n ) 1 v v . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ37_HTML.gif
(4.15)
Now taking w = 2 u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq228_HTML.gif and w = u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq229_HTML.gif in (4.14), respectively, we get that
Ω A n ( x ) ( u n ) u n d x = Ω u n ( v u n ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equo_HTML.gif
Using the estimate (4.9), (4.8) and Ω being a bounded subset in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq9_HTML.gif, we have that
Ω A 1 n ( x ) ( u n ) u n d x C 0 u n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equp_HTML.gif
By (4.15), it follows that
u n + 1 n Ω | u n | 2 d x Ω u n ( v u n ) d x C 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equq_HTML.gif
where C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq230_HTML.gif depends on v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq231_HTML.gif. Moreover, we obtain that for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq205_HTML.gif,
u n C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ38_HTML.gif
(4.16)
and
1 n Ω | u n | 2 d x C 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ39_HTML.gif
(4.17)

Thus, { u n } n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq232_HTML.gif is bounded in W 1 , 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq89_HTML.gif and we may extract a subsequence such that u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq233_HTML.gif converges in L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq79_HTML.gif. Now, by (4.15) and (4.16), we know that u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq234_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif and u BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq148_HTML.gif.

Observe that by (4.6) and (4.17), A n ( x ) ( u n ) = A n ( ) ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq235_HTML.gif is bounded in L 2 ( Ω ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq236_HTML.gif and weakly relatively compact in L 2 ( Ω ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq236_HTML.gif. We may assume that
A n ( ) ( u n ) z as  n  weakly in  L 2 ( Ω ; R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equr_HTML.gif
By (4.17) and 1 n | u n | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq237_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif, we also have that
A 1 n ( ) ( u n ) z as  n  weakly in  L 2 ( Ω ; R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ40_HTML.gif
(4.18)
Given φ C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq238_HTML.gif and taking w = u n ± φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq239_HTML.gif in (4.14), we have
Ω ( w u n ) φ d x = Ω A n ( x ) ( u n ) φ d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equs_HTML.gif
and letting n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq240_HTML.gif, it follows that
Ω ( w u ) φ d x = Ω z φ d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equt_HTML.gif
that is, v u = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq224_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq174_HTML.gif, and
div ( A 1 n ( ) ( u n ) ) div ( z ) as  n  weakly in  L 2 ( Ω ; R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ41_HTML.gif
(4.19)
From the proof of Proposition 4 in [11], we obtain that f ( x , z ( x ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq241_HTML.gif. Moreover, by (4.17) and (4.18), it implies that z M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq242_HTML.gif. Now, we prove that u, v and z verify (4.13). Applying the Lebesgue convergence theorem in (4.14), there exists u BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq148_HTML.gif for every w W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq243_HTML.gif,
Ω ( w u ) ( v u ) d x Ω z ( w u n ) d x Ω z w d x Ω z u d x = Ω z w d x Ω ( z , D u ) = Ω z w d x Ω | D u | f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ42_HTML.gif
(4.20)

To prove (4.13), we assume that there exists w 0 W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq244_HTML.gif. Let w W 1 , 1 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq153_HTML.gif and let w n W p 1 , 2 ( Ω ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq245_HTML.gif be such that w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq176_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq180_HTML.gif. Using w n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq179_HTML.gif as a test function in (4.20) and letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq180_HTML.gif, we obtain (4.13). That is, ( u , v u ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq246_HTML.gif.

Step 2. Now let us prove that D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq215_HTML.gif is dense in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif. We only need to prove that C 0 ( Ω ) D ( A ) ¯ L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq247_HTML.gif. Let v C 0 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq248_HTML.gif. By Step 1, we have that v R ( I + 1 n A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq249_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq250_HTML.gif. Thus, for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq205_HTML.gif, there exists u n D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq251_HTML.gif such that ( u n , v u n ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq252_HTML.gif and, in consequence, there exists some z n M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq253_HTML.gif such that n ( v u n ) = div ( z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq254_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq174_HTML.gif and
n Ω ( w u n ) ( v u n ) d x Ω z n w d x Ω | D u | f Ω z n w d x Ω | w | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equu_HTML.gif
for every w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq169_HTML.gif. Taking w = v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq255_HTML.gif in the above inequality, we get
Ω ( v u n ) 2 d x C n Ω | v | d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equv_HTML.gif

Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq256_HTML.gif, it follows that u n v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq257_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif. This implies that v D ( A ) ¯ L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq258_HTML.gif. □

Proof of Proposition 4.1 Let p P 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq207_HTML.gif and ( u , v ) , ( u ˜ , v ˜ ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq259_HTML.gif. Let z , z ˜ M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq260_HTML.gif be such that v = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq261_HTML.gif, v ˜ = div ( z ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq262_HTML.gif and
Ω ( w u ) v d x Ω ( z , D w ) Ω | D u | f , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ43_HTML.gif
(4.21)
Ω ( w u ˜ ) v ˜ d x Ω ( z ˜ , D w ) Ω | D u ˜ | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ44_HTML.gif
(4.22)
for every w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq169_HTML.gif. Taking w = u p ( u u ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq263_HTML.gif as a test function in (4.21), taking w = u ˜ + p ( u u ˜ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq264_HTML.gif as a test function in (4.22), and by Theorem 2 in [4], we have that
Ω p ( u u ˜ ) ( v v ˜ ) d x Ω ( z z ˜ , D p ( u u ˜ ) ) = Ω θ ( z z ˜ , D p ( u u ˜ ) , x ) | D p ( u u ˜ ) | 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equw_HTML.gif

We get the operator A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif is completely accretive in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif.

Now, we prove that A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif is closed. Let ( u n , v n ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq265_HTML.gif such that ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq266_HTML.gif in L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq267_HTML.gif. Since ( u n , v n ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq268_HTML.gif, there exists z n M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq269_HTML.gif with v n = div ( z n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq270_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq174_HTML.gif such that
Ω ( w u n ) v n d x Ω ( z n , D w ) Ω | D u n | f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ45_HTML.gif
(4.23)
for every w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq169_HTML.gif. Since z n C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq271_HTML.gif, we may assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ46_HTML.gif
(4.24)
Working as before, it is easy to see that z M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq242_HTML.gif. Moreover, since v n v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq272_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif, we have v = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq273_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq152_HTML.gif, and
lim n Ω ( z n , D w ) = Ω ( z , D w ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equx_HTML.gif
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq180_HTML.gif in (4.23), and having in mind the lower semicontinuity of the functional defined in (2.13), we obtain that
Ω ( w u ) v d x Ω ( z , D w ) Ω | D u | f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equy_HTML.gif
Consequently, ( u , v ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq274_HTML.gif. By Lemma 4.5, it follows that A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq275_HTML.gif is m-completely accretive in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif. By Crandall-Liggett’s theorem, A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq275_HTML.gif generates a contraction semigroup in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif given by the exponential formula
e t A u 0 = lim n ( I + t n A ) n u 0 for any  u 0 L 2 ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equz_HTML.gif
The function u ( t ) = e t A u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq276_HTML.gif is a mild solution of
d u d t + A u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equaa_HTML.gif

with u ( 0 ) = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq137_HTML.gif. □

Proof of Proposition 4.2 We first prove that A F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq277_HTML.gif. Let ( u , v ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq274_HTML.gif and w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq169_HTML.gif, there exists z M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq242_HTML.gif with v = div ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq278_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq152_HTML.gif such that
Ω ( w u ) v d x Ω ( z , D w ) Ω | D u | f F ( w ) F ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equab_HTML.gif

for every w BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq169_HTML.gif. Thus, ( u , v ) F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq279_HTML.gif, that is, A F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq280_HTML.gif.

Next, by the proof of Proposition 4.1, we have that the operator A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif is closed. Since A F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq281_HTML.gif and L ( Ω ) R ( I + A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq282_HTML.gif, we have that F = A ¯ L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq283_HTML.gif. □

Proof of Theorem 3.3 As a consequence of Proposition 4.2, the semigroups generated by A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq146_HTML.gif and by F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq156_HTML.gif coincide, and therefore u ( t , x ) : = e t A u 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq284_HTML.gif is a strong solution of
d u ( t ) d t + A u ( t ) 0 , in  L 2 ( Ω ) , t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equac_HTML.gif

with u ( 0 ) = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq137_HTML.gif, i.e., u W loc 1 , 2 ( [ 0 , T ] ; L 2 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq285_HTML.gif and ( u ( t ) , u ( t ) ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq286_HTML.gif for a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq287_HTML.gif. Then we have u ( t ) = div ( z ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq288_HTML.gif in D ( [ 0 , T ] × Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq105_HTML.gif. By the characterization (i) in Lemma 4.3, we have (3.2) and (3.3) hold. The contractivity estimate (3.5) follows directly from the nonlinear semigroup theory. □

Let us define several operators that will be needed in this section. The single-valued operator B h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq289_HTML.gif is defined in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif as
B h : = { ( u , v ) L 2 ( Ω ) × L 2 ( Ω ) : v = h ( , u ( ) )  a.e. } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equad_HTML.gif
Take H : L 2 ( Ω ) [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq290_HTML.gif defined by
H ( u ) : = { Ω k ( x , u ( x ) ) d x , x k ( x , u ( x ) ) L 1 ( Ω ) , + , otherwise, https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equae_HTML.gif

where k ( x , u ( x ) ) : = 0 r h ( x , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq291_HTML.gif. It is easy to see that L ( Ω ) D ( H ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq292_HTML.gif and H is convex. Moreover, by Fatou’s lemma, H is lower semicontinuous. Hence, ∂H is a maximal monotone graph in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq1_HTML.gif.

Thus, to prove Theorem 3.4, we only need to obtain the following result.

Lemma 4.6 ( H + F ) = A + B h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq293_HTML.gif.

Proof From ( u , v ) A + B h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq294_HTML.gif, we have u BV ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq148_HTML.gif, v L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq149_HTML.gif and there exists z M f 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq173_HTML.gif, v = div ( z ) + h ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq295_HTML.gif in D ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq174_HTML.gif such that
Ω ( w u ) v d x Ω ( z , D w ) d x Ω | D u | f + Ω ( w u ) h ( x , u ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equ47_HTML.gif
(4.25)
for all w W 1 , 1 ( Ω ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq296_HTML.gif. Since h ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq31_HTML.gif satisfies (M1), there exists z ( x ) [ u ( x ) , w ( x ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq297_HTML.gif such that
Ω u v h ( x , s ) d s d x = Ω ( w u ) h ( x , z ( x ) ) d x Ω ( w u ) h ( x , u ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equaf_HTML.gif
By the above inequality and being lower semicontinuous in (4.25), we have that
Ω ( w u ) v d x Ω ( z , D w ) d x Ω | D u | f + Ω ( w u ) h ( x , u ) d x Ω | D w | f Ω | D u | f + Ω u ( x ) v ( x ) h ( x , s ) d s d x = ( H + F ) ( w ) ( H + F ) ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_Equag_HTML.gif

We have proved that A + B h ( H + F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq298_HTML.gif.

By Proposition 14 in [19], we have that L ( Ω ) R ( I + A + B h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq299_HTML.gif, and the operator A + B h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq300_HTML.gif is closed. Hence, ( H + F ) = A + B h ¯ L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-147/MediaObjects/13661_2012_Article_402_IEq301_HTML.gif. □

Using Crandall-Liggett’s theorem and a similar proof of Theorem 3.3 again, we obtain that Theorem 3.4 holds.

Declarations

Acknowledgements

We would like to thank the anonymous referees for their constructive comments, which were very helpful for improving this paper. The authors acknowledge the financial support of this research by the National Natural Science Foundation of China (Grant No. 10871117), NSFSP (Grant No. ZR2010AM013) and Fundamental Research Funds for the Central Universities (12CX04081A, 11CX04058A).

Authors’ Affiliations

(1)
School of Mathematics, Shandong University
(2)
College of Science, China University of Petroleum (East China)

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© Rui and Si; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.