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Volume 2010, Article ID 208085, 23 pagesdoi:10.1155/2010/208085 Research Article The Boundary Value Problem of the Equations with Nonnegative Characteristic Form Limei Li and Tian Ma Mat

Volume 2010, Article ID 208085, 23 pages

doi:10.1155/2010/208085

Research Article

The Boundary Value Problem of the Equations with Nonnegative Characteristic Form

Limei Li and Tian Ma

Mathematical College, Sichuan University, Chengdu 610064, China

Correspondence should be addressed to Limei Li,matlilm@yahoo.cnand

Tian Ma,matian56@sina.com

Received 22 May 2010; Accepted 7 July 2010

Academic Editor: Claudianor Alves

Copyrightq 2010 L Li and T Ma This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

We study the generalized Keldys-Fichera boundary value problem for a class of higher order tions with nonnegative characteristic By using the acute angle principle and the H ¨older inequali-ties and Young inequalities we discuss the existence of the weak solution Then by using the inverse

equa-H ¨older inequalities, we obtain the regularity of the weak solution in the anisotropic Sobolev space

1 Introduction

Keldys1 studies the boundary problem for linear elliptic equations with degenerationg onthe boundary For the linear elliptic equations with nonnegative characteristic forms, Oleinikand Radkevich2 had discussed the Keldys-Fichera boundary value problem In 1989, Maand Yu3 studied the existence of weak solution for the Keldys-Fichera boundary value ofthe 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 degenerateelliptic 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 ofhigher-order equations with nonnegative characteristic form We discuss the existence anduniqueness of weak solution by using the acute angle principle, then study the regularity ofsolution by using inverse H ¨older inequalities in the anisotropic Sobolev Space

We firstly study the following linear partial differential operator

where x ∈ Ω, Ω ⊂ R nis an open set, the coefficients of L are bounded measurable, and theleading term coefficients satisfy

k1b λ i λ j x · n k , −→n  n1, , n n  is the outward normal at ∂Ω {e i x} N m

i1and{h i x} N m−1

i1 are the eigenvalues of matrices Mx and Bx, respectively C B

ij x and C M ij x are orthogonal matrix satisfying

2 Formulation of the Boundary Value Problem

For second-order equations with nonnegative characteristic form, Keldys 1 and Ficherapresented a kind of boundary that is the Keldys-Fichera boundary value problem, with thatthe 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 ofboundary value condition, which is consistent with Dirichlet problem if the equations areelliptic, and coincident with Keldys-Fichera boundary value problem when the equations are

where Cx is the transposed matrix of Cx, {e i x} N m

i1 are the eigenvalues of Mx and {h i x} N m−1

i1 are the eigenvalues of Bx Denote by

For multiple indices α, β, α ≤ β means that α i ≤ β i , for all 1 ≤ i ≤ n Now let us consider the

following boundary value problem,

We can see that the item 2.13 of boundary value condition is determined by theleading term matrix 2.2, and 2.12 is defined by the odd term matrix 2.6 Moreover, if

the operator L is a not elliptic, then the operator

 ∂4u

∂x2

1∂x2 2

∂3u

∂x3 2

− Δu  f, x ∈ Ω ⊂ R2. 2.15

HereΩ  {x1, x2 ∈ R2 | 0 < x1 < 1, 0 < x2 < 1 } Let α1 {2, 0}, α2 {1, 1} α3 {0, 2} and

λ1 {1, 0}, λ2 {0, 1}, then the leading and odd term matrices of 2.15 respectively are

which implies that ∂u/∂x2is free onΓ  {x1, x2 ∈ ∂Ω | 0 < x1< 1, x2 0}.

boundary value conditions relating to the operator L may not be unique.

2.11 and 2.12 remain

Now we return to discuss the relations between the conditions 2.11–2.13 withDirichlet and Keldys-Fichera boundary value conditions.

It is easy to verify that the problem2.10–2.13 is the Dirichlet problem provided the

operator L being ellipticsee 11 In this case,M

i  ∂Ω for all 1 ≤ i ≤ N m Besides,2.13run over all 1 ≤ i ≤ N m and δ k j ≤ α i , moreover C B x is nondegenerate for any x ∈ ∂Ω Solving the system of equations, we get D α u|∂Ω  0, for all |α|  m − 1.

When m  1, namely, L is of second-order, the condition 2.12 is the form

We denote by X2the completion of X under the norm 2and by X1 the completion of X

with the following norm

Let u be a classical solution of2.10–2.13 Denote by

From the three equalities above we obtain2.30.

Let u ∈ X1 be a weak solution of2.10–2.13 Then the boundary value conditions

2.11 and 2.13 can be reflected by the space X1 In fact, we can show that if u ∈ X1, then u

which means that2.35 holds true Since X is dense in X1, for u ∈ X1given, let u k ∈ X and

Due to u ksatisfying2.36, hence u ∈ X1satisfies2.36 Thus 2.31 is verified

D γ u|M

γ  0, forγ   m − 1, 2.39

where M

γ  {x ∈ ∂Ω | n

i1a γ δ iγ δ i x · n i2 > 0 } In this case, the corresponding trace

embedding theorem can be set, and the boundary value condition2.13 is naturally satisfied

On the other hand, if the weak solution u of2.10–2.13 belongs to X1∩ W m,pΩ for some

It remains to verify the condition2.12 Let u0 ∈ X1∩ W m 1,2Ω satisfy 2.30 Since

Because the coefficients of L are sufficiently smooth, and C∞

i , one deduces that u0satisfies2.12 provided u0 ∈ X1∩ W m 1,2 Ω.

Finally, we discuss the well-posedness of the boundary value problem2.10–2.13

Let X be a linear space, and X1, X2be the completion of X, respectively, with the norm

1, 2 Suppose that X1is a reflexive Banach space and X2is a separable Banach space

Theorem 2.8 existence theorem Let Ω ⊂ R n be an arbitrary open set, f ∈ L2Ω and b αγ ∈

C1Ω If there exist a constant C > 0 and g ∈ L1Ω such that

bounded linear operator L : X1 → X2 ∗ Hence L is weakly continuoussee 3 From 2.42,

for u ∈ X we drive that

−

$

C i

Thus by H ¨older inequalitysee 13, we have

)

ByLemma 2.7, the theorem is proven

Theorem 2.9 uniqueness theorem Under the assumptions of Theorem 2.8 with g x  0 in

2.48 If the problem 2.10–2.13 has a weak solution in X1 ∩ W m,p Ω ∩ W m −1,q Ω1/p 

1/q  1, then such a solution is unique Moreover, if b αγ x  0 in L, for all |α|  m, |γ|  m − 1,

then the weak solution u ∈ X1of2.10–2.13 is unique.

2.30 holds for all v ∈ X1∩ W m,p ∩ W m −1,q Ω Hence Lu0, u0is well defined Let u1 ∈ X1∩

W m,p ∩W m −1,qΩ Then from 2.49 it follows that < Lu1−Lu0, u1−u0 >  0, we obtain u1 u0,which means that the solution of2.10–2.13 in X1∩ W m,p ∩ W m −1,qΩ is unique If all the

odd terms b αγ x of L, then 2.30 holds for all v ∈ X1, in the same fashion we known that theweak solution of2.10–2.13 in X1is 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 X1∩W m,p Ω∩W m −1,q Ω1/p1/q  1.

3 Existence of Higher-Order Quasilinear Equations

Given the quasilinear differential operator

We consider the following problem:

k, |α|≤k Ω to L p Ω, if q θis the largest number of the

exponent p in where D θ u ∈ L p Ω, for all u ∈ W p α

|α|≤kΩ, and the embedding is continuous.For example, whenΩ is bounded, the space X  {u ∈ L k Ω | k ≥ 1, D i u ∈ L2Ω, 1 ≤

i L2 L k is an anisotropic Sobolev space, and the critical

embedding exponents from X to L P Ω are q i  21 ≤ i ≤ n, and q0 max{k, 2n/n − 2}.

Suppose that the following hold

A1 The coefficients of the leading term of A satisfy one of the following two conditions:

A3 There are functions G i x, η i  0, 1, , n with G i x, 0  0, for all 1 ≤ i ≤ n, such

where f1∈ L1Ω, p0> 1, p λ > 1 or p λ  0, for all 1 ≤ |λ| ≤ m − 2.

A5 There is a constant c > 0 such that

m −1,|λ|≤m−1 Ω to L P Ω Let X be defined by 2.27 and X1be the completion of X

under the norm

and X2be the completion of X with the norm

Theorem 3.1 Under the conditions A1 –A5, if f ∈ L p0 

Ω, 1/p01/p0   1, then the problem

3.3 has a weak solution in X1.

Au, v defines a bounded mapping A : X1 → X2 ∗by the conditionA5

Let u ∈ X, by A2–A4, one can deduce that

−

$

C i

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

Example 3.2 We consider the boundary value problem of odd order equation as follows:

whereΩ is an unit ball in R2, seeFigure 2

The odd term matrix is

ApplyingTheorem 3.1, if f ∈ L 4/3Ω, then the problem 3.15–3.18 has a weak solution

u ∈ W 1,2Ω

4. Wm,p-Solutions of Degenerate Elliptic Equations

We start with an abstract regularity result which is useful for the existence problem of

W m,p Ω-solutions of degenerate quasilinear elliptic equations of order 2m Let X, X1, X2bethe spaces defined in Definition 2.6, and Y be a reflective Banach space, at the same time

Y → X1

Lemma 4.1 Under the hypotheses of Lemma 2.7 , there exists a sequence of {u n } ⊂ X, u n u0in

B1 The condition 3.6 holds, and there is a continuous function λx ≥ 0 on Ω such

where C is a constant, p0> 1, p λ > 1 or p λ  0 for 1 ≤ |λ| ≤ m − 1, f1∈ L1Ω

B4 The structure conditions are

Definition 4.2 u∈ 3X1is a weak solution of4.2–4.5, if for any v ∈ X2, the following equalityholds:

Theorem 4.3 Under the assumptions B1 –B4, if f ∈ L p0 

, then the problem and4.2–4.5 has a

weak solution u∈ 3X1 Moreover, if there is a real number δ ≥ 1, such that

$

then the weak solution u ∈ W m,pΩ ∩ 3X1, p  2δ/1  δ.

any u ∈ X X is as that inSection 3 with 3Au, u  0, we have

Due toB1 and B3 we have

From4.15 and 4.17, the estimates 4.12 follows This completes the proof.

Next, we consider a quasilinear equation

problem4.2–4.5 has a weak solution u ∈ W m,pΩ ∩ 3X1, p  2δ/1  δ.

The proof ofTheorem 4.4is parallel to that ofTheorem 4.3; here we omit the detail

Acknowledgment

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

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

Example 3.2 We consider the boundary value problem of odd order equation as follows:...  δ.

The proof ofTheorem 4.4is parallel to that ofTheorem 4.3; here we omit the detail

Acknowledgment

This project was supported by the National Natural...

We start with an abstract regularity result which is useful for the existence problem of

W m,p Ω-solutions of degenerate quasilinear elliptic equations of order 2m