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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ1_HTML.gif
(1.1)
where http://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 http://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
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ3_HTML.gif
(1.3)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ4_HTML.gif
(1.4)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ5_HTML.gif
(1.5)

with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq4_HTML.gif , where http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq7_HTML.gif . The odd order term coefficients http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq8_HTML.gif can be made into a matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq9_HTML.gif is the outward normal at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq10_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq12_HTML.gif are the eigenvalues of matrices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq14_HTML.gif , respectively. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq16_HTML.gif are orthogonal matrix satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ6_HTML.gif
(1.6)
The boundary sets are
http://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):
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ8_HTML.gif
(1.8)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq17_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ9_HTML.gif
(2.1)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq20_HTML.gif are bounded measurable functions, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq21_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq22_HTML.gif be a series of functions with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq24_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq25_HTML.gif into a row http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq26_HTML.gif , then http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ10_HTML.gif
(2.2)
Suppose that the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq29_HTML.gif is semipositive, that is,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ12_HTML.gif
(2.4)
where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq31_HTML.gif , we have
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ14_HTML.gif
(2.6)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq32_HTML.gif is the outward normal at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq33_HTML.gif . Let the following matrices be orthogonal:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ15_HTML.gif
(2.7)
satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ16_HTML.gif
(2.8)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq34_HTML.gif is the transposed matrix of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq35_HTML.gif are the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq37_HTML.gif are the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq38_HTML.gif . Denote by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ17_HTML.gif
(2.9)
For multiple indices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq39_HTML.gif means that http://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,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ18_HTML.gif
(2.10)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ19_HTML.gif
(2.11)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ20_HTML.gif
(2.12)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ21_HTML.gif
(2.13)

for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq42_HTML.gif , where http://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 http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ23_HTML.gif
(2.15)
Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq45_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq46_HTML.gif and http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ25_HTML.gif
(2.17)
We can see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq48_HTML.gif , and http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ27_HTML.gif
(2.19)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq50_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq51_HTML.gif . Since only http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq52_HTML.gif , hence we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ28_HTML.gif
(2.20)
however, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq54_HTML.gif , therefore,
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ30_HTML.gif
(2.22)
which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq55_HTML.gif is free on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq56_HTML.gif
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq57_HTML.gif and http://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 http://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 http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq62_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq63_HTML.gif . Besides, (2.13) run over all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq65_HTML.gif , moreover http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq66_HTML.gif is nondegenerate for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq68_HTML.gif .

When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq69_HTML.gif , namely, http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ31_HTML.gif
(2.23)
and (2.13) is
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ32_HTML.gif
(2.24)
Noticing
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ34_HTML.gif
(2.26)

It shows that when http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ35_HTML.gif
(2.27)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq72_HTML.gif is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ36_HTML.gif
(2.28)
We denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq73_HTML.gif the completion of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq74_HTML.gif under the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq75_HTML.gif and by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq76_HTML.gif the completion of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq77_HTML.gif with the following norm
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ37_HTML.gif
(2.29)

Definition 2.4.

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq79_HTML.gif , the following equality holds:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq80_HTML.gif are sufficiently smooth.

Let http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ40_HTML.gif
(2.32)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq82_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ41_HTML.gif
(2.33)
Because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq83_HTML.gif satisfies (2.12),
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq86_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq87_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ43_HTML.gif
(2.35)
Evidently, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq88_HTML.gif , we have
http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq90_HTML.gif is dense in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq91_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq92_HTML.gif given, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq94_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq95_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ46_HTML.gif
(2.38)

Due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq96_HTML.gif satisfying (2.36), hence http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ47_HTML.gif
(2.39)

where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq100_HTML.gif for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq102_HTML.gif satisfy (2.30). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq103_HTML.gif , hence we have
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq104_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ49_HTML.gif
(2.41)
Because the coefficients of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq105_HTML.gif are sufficiently smooth, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq106_HTML.gif is dense in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq108_HTML.gif . Therefore, due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq109_HTML.gif , we have
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ51_HTML.gif
(2.43)
Furthermore,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ53_HTML.gif
(2.45)

Noticing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq110_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq111_HTML.gif , one deduces that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq112_HTML.gif satisfies (2.12) provided http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq114_HTML.gif be a linear space, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq115_HTML.gif be the completion of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq116_HTML.gif , respectively, with the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq117_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq118_HTML.gif is a reflexive Banach space and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq121_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq122_HTML.gif , one has
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq124_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ55_HTML.gif
(2.47)

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

Theorem 2.8 (existence theorem).

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

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq132_HTML.gif is the component of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq133_HTML.gif corresponding to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq135_HTML.gif .

Proof.

Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq137_HTML.gif defines a bounded linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq138_HTML.gif . Hence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq140_HTML.gif we drive that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ57_HTML.gif
(2.49)
Hence we obtain
http://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
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq143_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq144_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq145_HTML.gif , then the weak solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq148_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq149_HTML.gif is well defined. Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq151_HTML.gif , we obtain http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq154_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq155_HTML.gif , then (2.30) holds for all http://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 http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ60_HTML.gif
(3.1)

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

Let http://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), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq162_HTML.gif , and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ61_HTML.gif
(3.2)

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

We consider the following problem:
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ63_HTML.gif
(3.4)
whose norm is
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ64_HTML.gif
(3.5)

when all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq166_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq167_HTML.gif , then the space is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq168_HTML.gif . http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq170_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq171_HTML.gif , if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq173_HTML.gif in where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq175_HTML.gif is bounded, the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq176_HTML.gif with norm http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq178_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq179_HTML.gif are http://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 http://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:

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

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

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

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

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

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

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

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

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq208_HTML.gif , we have
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq209_HTML.gif , if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq211_HTML.gif .

Proof.

Denote by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq213_HTML.gif defines a bounded mapping http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq214_HTML.gif by the condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq215_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq216_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq217_HTML.gif , one can deduce that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ72_HTML.gif
(3.13)
Noticing that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq218_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq219_HTML.gif , http://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
http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ74_HTML.gif
(3.15)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq222_HTML.gif is an unit ball in http://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
http://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
http://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
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq225_HTML.gif .
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Fig2_HTML.jpg

Figure 2

4. http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq228_HTML.gif . Let http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq232_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq233_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq234_HTML.gif . Furthermore, if, we can derive that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq235_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq236_HTML.gif is a constant, then the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq237_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq238_HTML.gif belongs to http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ78_HTML.gif
(4.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq241_HTML.gif . Now, we consider the following problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ79_HTML.gif
(4.2)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ80_HTML.gif
(4.3)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ81_HTML.gif
(4.4)
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq244_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq245_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ83_HTML.gif
(4.6)

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

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

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

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

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

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

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

Definition 4.2.

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq269_HTML.gif , the following equality holds:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq270_HTML.gif , if http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq273_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ88_HTML.gif
(4.11)

then the weak solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq275_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq276_HTML.gif ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq278_HTML.gif , we have
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ90_HTML.gif
(4.13)
Due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq279_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq280_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ91_HTML.gif
(4.14)
Noticing that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq281_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq282_HTML.gif consequently we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ92_HTML.gif
(4.15)
where the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq283_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq284_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq285_HTML.gif is the critical embedding exponent from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq286_HTML.gif to http://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])
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ93_HTML.gif
(4.16)
Then we obtain
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ95_HTML.gif
(4.18)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq290_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_Equ96_HTML.gif
(4.19)
The structural conditions are
http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq292_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq293_HTML.gif are the critical embedding exponents from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq294_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq295_HTML.gif .

Theorem 4.4.

Let the conditions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq296_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F208085/MediaObjects/13661_2010_Article_904_IEq297_HTML.gif be satisfied. If http://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 http://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

References

  1. Keldys MV: On certain cases of degeneration of equations of elliptic type on the boundry of a domain. Doklady Akademii Nauk SSSR 1951, 77: 181-183.MathSciNet
  2. Oleinik OA, Radkevich EV: Second Order Equation with Nonnegative Characteristic Form. Plennum Press, New York, NY, USA; 1973.View Article
  3. Ma T, Yu QY: The Keldys-Fichera boundary value problems for degenerate quasilinear elliptic equations of second order. Differential and Integral Equations 1989,2(4):379-388.MathSciNet
  4. Chen ZC: The Keldys-Fichera boundary value problem for a class of nonlinear degenerate elliptic equations. Acta Mathematica Sinica. New Series 1993,9(2):203-211. 10.1007/BF02560051MathSciNetView Article
  5. Chen Z-C, Xuan B-J: On the Keldys-Fichera boundary-value problem for degenerate quasilinear elliptic equations. Electronic Journal of Differential Equations 2002, (87):1-13.
  6. Li SH: The first boundary value problem for quasilinear elliptic-parabolic equations with double degenerate. Nonlinear Analysis: Theory, Methods & Applications 1996,27(1):115-124. 10.1016/0362-546X(94)00334-EMathSciNetView Article
  7. Wang L: Hölder estimates for subelliptic operators. Journal of Functional Analysis 2003,199(1):228-242. 10.1016/S0022-1236(03)00093-4MathSciNetView Article
  8. Li LM, Ma T: Regularity of Keldys-Fichera boundary value problem for gegenerate elliptic equations. Chinese Annals of Mathematics, Series B 2010,31(B(5–6)):1-10.MathSciNet
  9. Ma T: Weakly continuous method and nonlinear differential equations with nonnegative characteristic form, Doctoral thesis. 1989.
  10. Ma T, Yu QY: Nonlinear partial differential equations and weakly continuous methed. In Nonlinear Analysis. Edited by: Chen W. Lanzhou Univ. Press; 1990:118-213.
  11. Ladyzhenskaya OA, Uraltseva NN: Linear and Quasilinear Equations of Elliptic Type. 2nd edition. Izdat. "Nauka", Moscow, Russia; 1973:576.
  12. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 22. Springer, Berlin, Germany; 1977:x+401.View Article
  13. Temam R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences. Volume 68. Springer, New York, NY, USA; 1988:xvi+500.View Article
  14. Adams RA: Sobolev Spaces, Pure and Applied Mathematics, Vol. 6. Academic Press, London, UK; 1975:xviii+268.

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© Limei Li and Tian Ma. 2010

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