Open Access

The Boundary Value Problem of the Equations with Nonnegative Characteristic Form

Boundary Value Problems20102010:208085

DOI: 10.1155/2010/208085

Received: 22 May 2010

Accepted: 7 July 2010

Published: 25 July 2010

Abstract

We study the generalized Keldys-Fichera boundary value problem for a class of higher order equations with nonnegative characteristic. By using the acute angle principle and the Hölder inequalities and Young inequalities we discuss the existence of the weak solution. Then by using the inverse Hölder inequalities, we obtain the regularity of the weak solution in the anisotropic Sobolev space.

1. Introduction

Keldys [1] studies the boundary problem for linear elliptic equations with degenerationg on the boundary. For the linear elliptic equations with nonnegative characteristic forms, Oleinik and Radkevich [2] had discussed the Keldys-Fichera boundary value problem. In 1989, Ma and Yu [3] studied the existence of weak solution for the Keldys-Fichera boundary value of the nonlinear degenerate elliptic equations of second-order. Chen [4] and Chen and Xuan [5], Li [6], and Wang [7] had investigated the existence and the regularity of degenerate elliptic equations by using different methods. In this paper, we study the generalized Keldys-Fichera boundary value problem which is a kind of new boundary conditions for a class of higher-order equations with nonnegative characteristic form. We discuss the existence and uniqueness of weak solution by using the acute angle principle, then study the regularity of solution by using inverse Hölder inequalities in the anisotropic Sobolev Space.

We firstly study the following linear partial differential operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq1_HTML.gif is an open set, the coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq2_HTML.gif are bounded measurable, and the leading term coefficients satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ2_HTML.gif
(1.2)
We investigate the generalized Keldys-Fichera boundary value conditions as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ5_HTML.gif
(1.5)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq5_HTML.gif .

The leading term coefficients are symmetric, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq6_HTML.gif which can be made into a symmetric matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq7_HTML.gif . The odd order term coefficients https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq8_HTML.gif can be made into a matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq9_HTML.gif is the outward normal at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq10_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq12_HTML.gif are the eigenvalues of matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq13_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq14_HTML.gif , respectively. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq16_HTML.gif are orthogonal matrix satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ6_HTML.gif
(1.6)
The boundary sets are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ7_HTML.gif
(1.7)
At last, we study the existence and regularity of the following quasilinear differential operator with boundary conditions (1.3)–(1.5):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ8_HTML.gif
(1.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq18_HTML.gif

This paper is a generalization of [3, 810].

2. Formulation of the Boundary Value Problem

For second-order equations with nonnegative characteristic form, Keldys [1] and Fichera presented a kind of boundary that is the Keldys-Fichera boundary value problem, with that the associated problem is of well-posedness. However, for higher-order ones, the discussion of well-posed boundary value problem has not been seen. Here we will give a kind of boundary value condition, which is consistent with Dirichlet problem if the equations are elliptic, and coincident with Keldys-Fichera boundary value problem when the equations are of second-order.

We consider the linear partial differential operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ9_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq19_HTML.gif is an open set, the coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq20_HTML.gif are bounded measurable functions, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq21_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq22_HTML.gif be a series of functions with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq23_HTML.gif . If in certain order we put all multiple indexes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq24_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq25_HTML.gif into a row https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq26_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq27_HTML.gif can be made into a symmetric matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq28_HTML.gif . By this rule, we get a symmetric leading term matrix of (2.1), as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ10_HTML.gif
(2.2)
Suppose that the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq29_HTML.gif is semipositive, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ11_HTML.gif
(2.3)
and the odd order part of (2.1) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ12_HTML.gif
(2.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq30_HTML.gif is the Kronecker symbol. Assume that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq31_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ13_HTML.gif
(2.5)
We introduce another symmetric matrix
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ14_HTML.gif
(2.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq32_HTML.gif is the outward normal at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq33_HTML.gif . Let the following matrices be orthogonal:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ15_HTML.gif
(2.7)
satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ16_HTML.gif
(2.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq34_HTML.gif is the transposed matrix of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq35_HTML.gif are the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq37_HTML.gif are the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq38_HTML.gif . Denote by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ17_HTML.gif
(2.9)
For multiple indices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq39_HTML.gif means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq40_HTML.gif . Now let us consider the following boundary value problem,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ18_HTML.gif
(2.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ19_HTML.gif
(2.11)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ20_HTML.gif
(2.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ21_HTML.gif
(2.13)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq42_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq43_HTML.gif .

We can see that the item (2.13) of boundary value condition is determined by the leading term matrix (2.2), and (2.12) is defined by the odd term matrix (2.6). Moreover, if the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq44_HTML.gif is a not elliptic, then the operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ22_HTML.gif
(2.14)

has to be elliptic.

In order to illustrate the boundary value conditions (2.11)–(2.13), in the following we give an example.

Example 2.1.

Given the differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ23_HTML.gif
(2.15)
Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq45_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq47_HTML.gif , then the leading and odd term matrices of (2.15) respectively are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ24_HTML.gif
(2.16)
and the orthogonal matrices are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ25_HTML.gif
(2.17)
We can see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq48_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq49_HTML.gif as shown in Figure 1.The item (2.12) is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ26_HTML.gif
(2.18)
and the item (2.13) is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ27_HTML.gif
(2.19)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq51_HTML.gif . Since only https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq52_HTML.gif , hence we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ28_HTML.gif
(2.20)
however, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq54_HTML.gif , therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ29_HTML.gif
(2.21)
Thus the associated boundary value condition of (2.15) is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ30_HTML.gif
(2.22)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq55_HTML.gif is free on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq56_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Fig1_HTML.jpg

Figure 1

Remark 2.2.

In general the matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq58_HTML.gif arranged are not unique, hence the boundary value conditions relating to the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq59_HTML.gif may not be unique.

Remark 2.3.

When all leading terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq60_HTML.gif are zero, (2.10) is an odd order one. In this case, only (2.11) and (2.12) remain.

Now we return to discuss the relations between the conditions (2.11)–(2.13) with Dirichlet and Keldys-Fichera boundary value conditions.

It is easy to verify that the problem (2.10)–(2.13) is the Dirichlet problem provided the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq61_HTML.gif being elliptic (see [11]). In this case, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq62_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq63_HTML.gif . Besides, (2.13) run over all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq65_HTML.gif , moreover https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq66_HTML.gif is nondegenerate for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq67_HTML.gif . Solving the system of equations, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq68_HTML.gif .

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq69_HTML.gif , namely, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq70_HTML.gif is of second-order, the condition (2.12) is the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ31_HTML.gif
(2.23)
and (2.13) is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ32_HTML.gif
(2.24)
Noticing
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ33_HTML.gif
(2.25)
thus the condition (2.13) is the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ34_HTML.gif
(2.26)

It shows that when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq71_HTML.gif , (2.12) and (2.13) are coincide with Keldys-Fichera boundary value condition.

Next, we will give the definition of weak solutions of (2.10)–(2.13) (see [12]). Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ35_HTML.gif
(2.27)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq72_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ36_HTML.gif
(2.28)
We denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq73_HTML.gif the completion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq74_HTML.gif under the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq75_HTML.gif and by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq76_HTML.gif the completion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq77_HTML.gif with the following norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ37_HTML.gif
(2.29)

Definition 2.4.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq78_HTML.gif is a weak solution of (2.10)–(2.13) if for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq79_HTML.gif , the following equality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ38_HTML.gif
(2.30)

We need to check the reasonableness of the boundary value problem (2.10)–(2.13) under the definition of weak solutions, that is, the solution in the classical sense are necessarily the solutions in weak sense, and conversely when a weak solution satisfies certain regularity conditions, it will surely satisfy the given boundary value conditions. Here, we assume that all coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq80_HTML.gif are sufficiently smooth.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq81_HTML.gif be a classical solution of (2.10)–(2.13). Denote by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ39_HTML.gif
(2.31)
Thanks to integration by part, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ40_HTML.gif
(2.32)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq82_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ41_HTML.gif
(2.33)
Because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq83_HTML.gif satisfies (2.12),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ42_HTML.gif
(2.34)

From the three equalities above we obtain (2.30).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq84_HTML.gif be a weak solution of (2.10)–(2.13). Then the boundary value conditions (2.11) and (2.13) can be reflected by the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq85_HTML.gif . In fact, we can show that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq86_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq87_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ43_HTML.gif
(2.35)
Evidently, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq88_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ44_HTML.gif
(2.36)
If we can verify that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq89_HTML.gif , (2.36) holds true, then we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ45_HTML.gif
(2.37)
which means that (2.35) holds true. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq90_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq91_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq92_HTML.gif given, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq94_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq95_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ46_HTML.gif
(2.38)

Due to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq96_HTML.gif satisfying (2.36), hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq97_HTML.gif satisfies (2.36). Thus (2.31) is verified.

Remark 2.5.

When (2.2) is a diagonal matrix, then (2.13) is the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ47_HTML.gif
(2.39)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq98_HTML.gif In this case, the corresponding trace embedding theorem can be set, and the boundary value condition (2.13) is naturally satisfied. On the other hand, if the weak solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq99_HTML.gif of (2.10)–(2.13) belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq100_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq101_HTML.gif , then by the trace embedding theorems, the condition (2.13) also holds true.

It remains to verify the condition (2.12). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq102_HTML.gif satisfy (2.30). Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq103_HTML.gif , hence we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ48_HTML.gif
(2.40)
On the other hand, by (2.30), for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq104_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ49_HTML.gif
(2.41)
Because the coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq105_HTML.gif are sufficiently smooth, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq106_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq107_HTML.gif , equality (2.41) also holds for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq108_HTML.gif . Therefore, due to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq109_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ50_HTML.gif
(2.42)
From (2.36), one drives
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ51_HTML.gif
(2.43)
Furthermore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ52_HTML.gif
(2.44)
From (2.30) and (2.42), one can see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ53_HTML.gif
(2.45)

Noticing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq110_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq111_HTML.gif , one deduces that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq112_HTML.gif satisfies (2.12) provided https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq113_HTML.gif Finally, we discuss the well-posedness of the boundary value problem (2.10)–(2.13).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq114_HTML.gif be a linear space, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq115_HTML.gif be the completion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq116_HTML.gif , respectively, with the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq117_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq118_HTML.gif is a reflexive Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq119_HTML.gif is a separable Banach space.

Definition 2.6.

A mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq120_HTML.gif is called to be weakly continuous, if for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq121_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq122_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ54_HTML.gif
(2.46)

Lemma 2.7 (see [3]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq123_HTML.gif is a weakly continuous, if there exists a bounded open set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq124_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ55_HTML.gif
(2.47)

then the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq125_HTML.gif has a solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq126_HTML.gif .

Theorem 2.8 (existence theorem).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq127_HTML.gif be an arbitrary open set, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq129_HTML.gif . If there exist a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq131_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ56_HTML.gif
(2.48)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq132_HTML.gif is the component of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq133_HTML.gif corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq134_HTML.gif , then the problem (2.10)–(2.13) has a weak solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq135_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq136_HTML.gif be the inner product as in (2.31). It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq137_HTML.gif defines a bounded linear operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq138_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq139_HTML.gif is weakly continuous (see [3]). From (2.42), for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq140_HTML.gif we drive that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ57_HTML.gif
(2.49)
Hence we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ58_HTML.gif
(2.50)
Thus by Hölder inequality (see [13]), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ59_HTML.gif
(2.51)

By Lemma 2.7, the theorem is proven.

Theorem 2.9 (uniqueness theorem).

Under the assumptions of Theorem 2.8 with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq141_HTML.gif in (2.48). If the problem (2.10)–(2.13) has a weak solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq142_HTML.gif , then such a solution is unique. Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq143_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq144_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq145_HTML.gif , then the weak solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq146_HTML.gif of (2.10)–(2.13) is unique.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq147_HTML.gif be a weak solution of (2.10)–(2.13). We can see that (2.30) holds for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq148_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq149_HTML.gif is well defined. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq150_HTML.gif . Then from (2.49) it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq151_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq152_HTML.gif , which means that the solution of (2.10)–(2.13) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq153_HTML.gif is unique. If all the odd terms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq154_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq155_HTML.gif , then (2.30) holds for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq156_HTML.gif , in the same fashion we known that the weak solution of (2.10)–(2.13) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq157_HTML.gif is unique. The proof is complete.

Remark 2.10.

In next subsection, we can see that under certain assumptions, the weak solutions of degenerate elliptic equations are in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq158_HTML.gif .

3. Existence of Higher-Order Quasilinear Equations

Given the quasilinear differential operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ60_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq160_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq161_HTML.gif , the odd order part of (3.1) be as that in (2.4), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq162_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq163_HTML.gif be the same as those in Section 2. The leading matrix is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ61_HTML.gif
(3.2)

and the eigenvalues are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq164_HTML.gif . We denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq165_HTML.gif

We consider the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ62_HTML.gif
(3.3)
Denote the anisotropic Sobolev space by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ63_HTML.gif
(3.4)
whose norm is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ64_HTML.gif
(3.5)

when all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq166_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq167_HTML.gif , then the space is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq168_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq169_HTML.gif is termed the critical embedding exponent from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq170_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq171_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq172_HTML.gif is the largest number of the exponent https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq173_HTML.gif in where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq174_HTML.gif , and the embedding is continuous.

For example, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq175_HTML.gif is bounded, the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq176_HTML.gif with norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq177_HTML.gif is an anisotropic Sobolev space, and the critical embedding exponents from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq178_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq179_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq180_HTML.gif .

Suppose that the following hold.

The coefficients of the leading term of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq182_HTML.gif satisfy one of the following two conditions:

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq184_HTML.gif

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq186_HTML.gif

There is a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq188_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ65_HTML.gif
(3.6)
There are functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq190_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq191_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ66_HTML.gif
(3.7)
There is a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq193_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ67_HTML.gif
(3.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq194_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq195_HTML.gif .

There is a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq197_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ68_HTML.gif
(3.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq198_HTML.gif is a critical embedding exponent from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq199_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq200_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq201_HTML.gif be defined by (2.27) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq202_HTML.gif be the completion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq203_HTML.gif under the norm

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ69_HTML.gif
(3.10)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq204_HTML.gif be the completion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq205_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ70_HTML.gif
(3.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq206_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq207_HTML.gif is a weak solution of (3.3), if for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq208_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ71_HTML.gif
(3.12)

Theorem 3.1.

Under the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq209_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq210_HTML.gif , then the problem (3.3) has a weak solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq211_HTML.gif .

Proof.

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq212_HTML.gif the left part of (3.12). It is easy to verify that the inner product https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq213_HTML.gif defines a bounded mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq214_HTML.gif by the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq215_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq216_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq217_HTML.gif , one can deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ72_HTML.gif
(3.13)
Noticing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq220_HTML.gif , by Hölder and Young inequalities (see[13]), from (3.13) we can get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ73_HTML.gif
(3.14)

Ones can easily show that the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq221_HTML.gif is weakly continuous. Here we omit the details of the proof. By Lemma 2.7, this theorem is proven.

In the following, we take an example to illustrate the application of Theorem 3.1.

Example 3.2.

We consider the boundary value problem of odd order equation as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ74_HTML.gif
(3.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq222_HTML.gif is an unit ball in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq223_HTML.gif , see Figure 2

The odd term matrix is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ75_HTML.gif
(3.16)
It is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ76_HTML.gif
(3.17)
The boundary value condition associated with (3.15) is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ77_HTML.gif
(3.18)
Applying Theorem 3.1, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq224_HTML.gif , then the problem (3.15)–(3.18) has a weak solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq225_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Fig2_HTML.jpg

Figure 2

4. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq226_HTML.gif -Solutions of Degenerate Elliptic Equations

We start with an abstract regularity result which is useful for the existence problem of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq227_HTML.gif -solutions of degenerate quasilinear elliptic equations of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq228_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq229_HTML.gif be the spaces defined in Definition 2.6, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq230_HTML.gif be a reflective Banach space, at the same time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq231_HTML.gif .

Lemma 4.1.

Under the hypotheses of Lemma 2.7, there exists a sequence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq232_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq233_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq234_HTML.gif . Furthermore, if, we can derive that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq236_HTML.gif is a constant, then the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq237_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq238_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq239_HTML.gif .

In the following, we give some existence theorems of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq240_HTML.gif -solutions for the boundary value conditions (4.3)–(4.5) of higher-order degenerate elliptic equations.

First, we consider the quasilinear equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ78_HTML.gif
(4.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq241_HTML.gif . Now, we consider the following problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ79_HTML.gif
(4.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ80_HTML.gif
(4.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ81_HTML.gif
(4.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ82_HTML.gif
(4.5)

The boundary value condition associated with (4.1) is given by (4.3)–(4.5). Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq242_HTML.gif is bounded, and the following assumptions hold.

The condition (3.6) holds, and there is a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq244_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq245_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ83_HTML.gif
(4.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq247_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq249_HTML.gif is a measure zero set in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq250_HTML.gif , and there is a sequence of subdomains https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq251_HTML.gif with cone property such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq252_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq253_HTML.gif

The positive definite condition is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ84_HTML.gif
(4.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq255_HTML.gif is a constant, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq256_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq257_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq258_HTML.gif .

The structure conditions are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ85_HTML.gif
(4.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq260_HTML.gif is a constant, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq261_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq262_HTML.gif is the critical embedding exponent from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq263_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq264_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq265_HTML.gif be defined by (2.27) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq266_HTML.gif be the completion of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq267_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ86_HTML.gif
(4.9)

Definition 4.2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq268_HTML.gif is a weak solution of (4.2)–(4.5), if for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq269_HTML.gif , the following equality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ87_HTML.gif
(4.10)

Theorem 4.3.

Under the assumptions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq270_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq271_HTML.gif , then the problem and (4.2)–(4.5) has a weak solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq272_HTML.gif . Moreover, if there is a real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq273_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ88_HTML.gif
(4.11)

then the weak solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq274_HTML.gif

Proof.

According to Lemma 4.1, it suffices to prove that there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq275_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq276_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq277_HTML.gif is as that in Section 3) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq278_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ89_HTML.gif
(4.12)
From (4.10) we know
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ90_HTML.gif
(4.13)
Due to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq279_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq280_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ91_HTML.gif
(4.14)
Noticing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq281_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq282_HTML.gif consequently we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ92_HTML.gif
(4.15)
where the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq283_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq284_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq285_HTML.gif is the critical embedding exponent from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq286_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq287_HTML.gif . By the reversed Hölder inequality (see [14])
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ93_HTML.gif
(4.16)
Then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ94_HTML.gif
(4.17)

From (4.15) and (4.17), the estimates (4.12) follows. This completes the proof.

Next, we consider a quasilinear equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ95_HTML.gif
(4.18)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq288_HTML.gif

Suppose that the following holds.

There is a real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq290_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ96_HTML.gif
(4.19)
The structural conditions are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ97_HTML.gif
(4.20)

where C is a constant, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq292_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq293_HTML.gif are the critical embedding exponents from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq294_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq295_HTML.gif .

Theorem 4.4.

Let the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq297_HTML.gif be satisfied. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq298_HTML.gif , then the problem (4.2)–(4.5) has a weak solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq299_HTML.gif .

The proof of Theorem 4.4 is parallel to that of Theorem 4.3; here we omit the detail.

Declarations

Acknowledgment

This project was supported by the National Natural Science Foundation of China (no. 10971148).

Authors’ Affiliations

(1)
Mathematical College, Sichuan University

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© Limei Li and Tian Ma. 2010

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