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Hindawi Publishing Corporation Boundary Value Problems 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.cn and Tian Ma, matian56@sina.com Received 22 May 2010; Accepted 7 July 2010 Academic Editor: Claudianor Alves Copyright q 2010 L. Li and T. Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the generalized Keldys-Fichera boundary value problem for a class of higher order equa- tions with nonnegative characteristic. By using the acute angle principle and the H ¨ older inequali- ties andYounginequalities we discuss the existence of the weak solution. Then by using the inverse H ¨ older 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 H ¨ older inequalities in the anisotropic Sobolev Space. We firstly study the following linear partial differential operator Lu   | α |  | β | m, | γ | m−1  −1  m D α  a αβ  x  D β u  b αγ  x  D γ u    | θ | , | λ | ≤m−1  −1  |θ| D θ  d θλ  x  D λ u  , 1.1 2 Boundary Value Problems where x ∈ Ω, Ω ⊂ R n is an open set, the coefficients of L are bounded measurable, and the leading term coefficients satisfy a αβ  x  ξ α ξ β ≥ 0. 1.2 We investigate the generalized Keldys-Fichera boundary value conditions as follows: D α u| ∂Ω  0, | α | ≤ m − 2, 1.3 N m−1  j1 C B ij  x  D λ j u|  B i  0,    λ j     m − 1, 1 ≤ i ≤ N m−1 , 1.4 N m  j1 C M ij  x  D α j −δ k j u · n k j |  M i  0, ∀δ k j ≤ α j , 1.5 with |α j |  m and 1 ≤ i ≤ N m , where δ k j  {0, ,1   k j , ,0}. The leading term coefficients are symmetric, that is, a αβ xa βα x which can be made into a symmetric matrix Mxa α i α j . The odd order term coefficients b θλ x can be made into a matrix Bx  n k1 b λ i λ j x · n k , −→ n n 1 , ,n n  is the outward normal at ∂Ω. {e i x} N m i1 and {h i x} N m−1 i1 are the eigenvalues of matrices Mx and Bx, respectively. C B ij x and C M ij x are orthogonal matrix satisfying C M ij  x  M  x  C M ij  x     e i xδ ij  i,j1, ,N m , C B ij  x  B  x  C B ij  x     h i xδ ij  i,j1, ,N m−1 . 1.6 The boundary sets are M  i  { x ∈ ∂Ω | e i  x  > 0 } , 1 ≤ i ≤ N m , B  i  { x ∈ ∂Ω | h i  x  > 0 } , 1 ≤ i ≤ N m−1 . 1.7 At last, we study the existence and regularity of the following quasilinear differential operator with boundary conditions 1.3–1.5: Au   | α |  | β | m, | γ | m−1  −1  m D α  a αβ  x,  u  D β u  b αγ  x  D γ u    | γ |  | θ | m−1  −1  m−1 D γ  d γθ  x,  u  D θ u    | λ | ≤m−1  −1  |λ| D λ g λ  x,  u  , 1.8 where m ≥ 2and  u  {D α u} |α|≤m−2 . This paper is a generalization of 3, 8–10. Boundary Value Problems 3 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 Lu   | α |  | β | m, | γ | m−1  −1  m D α  a αβ  x  D β u  b αγ  x  D γ u    | θ | , | λ | ≤m−1  −1  |θ| D θ  d θλ  x  D λ u  , 2.1 where x ∈ Ω, Ω ⊂ R n is an open set, the coefficients of L are bounded measurable functions, and a αβ xa βα x. Let {g αβ x} be a series of functions with g αβ  g βα , |α|  |β|  k. If in certain order we put all multiple indexes α with |α|  k into a row {α 1 , ,α N k }, then {g αβ x} can be made into a symmetric matrix g α i α j . By this rule, we get a symmetric leading term matrix of 2.1, as follows: M  x    a α i α j x  i,j1, ,N m . 2.2 Suppose that the matrix Mx is semipositive, that is, 0 ≤ a α i α j  x  ξ i ξ j , ∀x ∈ Ω,ξ∈ R N m , 2.3 and the odd order part of 2.1 can be written as  | α | m, | γ | m−1  −1  m D α  b αγ  x  D γ u   n  i1  | λ |  | θ | m−1  −1  m D λδ i  b i λθ  x  D θ u  , 2.4 where δ i  {δ i1 , ,δ in },δ ij is the Kronecker symbol. Assume that for all 1 ≤ i ≤ n, we have b i λθ  x   b i θλ  x  ,x∈ Ω. 2.5 We introduce another symmetric matrix B  x    n  k1 b k λ i λ j x · n k  i,j1, ,N m−1 ,x∈ ∂Ω, 2.6 4 Boundary Value Problems where −→ n  {n 1 ,n 2 , ,n n } is the outward normal at x ∈ ∂Ω. Let the following matrices be orthogonal: C M  x    C M ij x  i,j1, ,N m ,x∈ Ω, C B  x    C B ij x  i,j1, ,N m−1 ,x∈ ∂Ω, 2.7 satisfying C M  x  M  x  C M  x     e i xδ ij  i,j1, ,N m , C B  x  B  x  C B  x     h i xδ ij  i,j1, ,N m−1 , 2.8 where Cx  is the transposed matrix of Cx, {e i x} N m i1 are the eigenvalues of Mx and {h i x} N m−1 i1 are the eigenvalues of Bx. Denote by M  i  { x ∈ ∂Ω | e i  x  > 0 } , 1 ≤ i ≤ N m , B  i  { x ∈ ∂Ω | h i  x  > 0 } , 1 ≤ i ≤ N m−1 , C  i  ∂Ω \ B  i , 1 ≤ i ≤ N m−1 . 2.9 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, Lu  f  x  ,x∈ Ω, 2.10 D α u| ∂Ω  0, | α | ≤ m − 2, 2.11 N m−1  j1 C B ij  x  D λ j u|  B i  0,    λ j     m − 1, 1 ≤ i ≤ N m−1 , 2.12 N m  j1 C M ij  x  D α j −δ k j u · n k j |  M i  0, 2.13 for all δ k j ≤ α j , |α j |  m and 1 ≤ i ≤ N m , where δ k j  {0, ,1   k j , ,0}. Boundary Value Problems 5 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 L is a not elliptic, then the operator L  u   | θ | , | λ | ≤m−1  −1  |θ| D θ  d θλ  x  D λ u  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 ∂ 4 u ∂x 4 1  ∂ 4 u ∂x 2 1 ∂x 2 2  ∂ 3 u ∂x 3 2 − Δu  f, x ∈ Ω ⊂ R 2 . 2.15 Here Ω{x 1 ,x 2  ∈ R 2 | 0 <x 1 < 1, 0 <x 2 < 1}.Letα 1  {2, 0},α 2  {1, 1}.α 3  {0, 2} and λ 1  {1, 0},λ 2  {0, 1}, then the leading and odd term matrices of 2.15 respectively are M  ⎛ ⎜ ⎜ ⎝ 100 010 000 ⎞ ⎟ ⎟ ⎠ , B   00 0 n 2  , 2.16 and the orthogonal matrices are C M  ⎛ ⎜ ⎜ ⎝ 100 010 001 ⎞ ⎟ ⎟ ⎠ , C B   10 01  . 2.17 We can see that  M 1  ∂Ω,  M 2  ∂Ω,  M 3  φ,and  B 1  φ,  B 2 as shown in Figure 1. The item 2.12 is 2  j1 C B 2j D λ j u|  B 2  D λ 2 u|  B 2  ∂u ∂x 2      B 2  0, 2.18 6 Boundary Value Problems x 2 x 1 Σ B 2 Γ Ω Figure 1 and the item 2.13 is 3  j1 C M 1j D α j −δ k j u · n k j |  M 1  D α 1 −δ k 1 u · n k 1 |  M 1  0, 3  j1 C M 2j D α j −δ k j u · n k j |  M 2  D α 2 −δ k 2 u · n k 2 |  M 2  0, 2.19 for all δ k 1 ≤ α 1 and δ k 2 ≤ α 2 . Since only δ k 1  {1, 0}≤α 1  {2, 0}, hence we have D α 1 −δ k 1 u · n k 1 |  M 1  ∂u ∂x 1 · n 1 | ∂Ω  0, 2.20 however, δ k 2  {1, 0} <α 2  {1, 1} and δ k 2  {0, 1} <α 2 , therefore, D α 2 −δ k 2 u · n k 2 |  M 2  ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∂u ∂x 2 · n 1 | ∂Ω  0, ∂u ∂x 1 · n 2 | ∂Ω  0. 2.21 Thus the associated boundary value condition of 2.15 is as follows: u| ∂Ω  0, ∂u ∂x 2     ∂Ω/Γ  0, ∂u ∂x 1     ∂Ω  0, 2.22 which implies that ∂u/∂x 2 is free on Γ{x 1 ,x 2  ∈ ∂Ω | 0 <x 1 < 1,x 2  0}. Remark 2.2. In general the matrices Mx and Bx arranged are not unique, hence the boundary value conditions relating to the operator L may not be unique. Remark 2.3. When all leading terms of L are zero, 2.10 is an odd order one. In this case, only 2.11 and 2.12 remain. Boundary Value Problems 7 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 L being elliptic see 11. In this case,  M i  ∂Ω for all 1 ≤ i ≤ N m . Besides, 2.13 run 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 u|  B  0, B    x ∈ ∂Ω | n  i1 b i  x  n i > 0  , 2.23 and 2.13 is n  j1 C M ij  x  n j u|  M i  0, 1 ≤ i ≤ n. 2.24 Noticing n  i,j1 a ij  x  n i n j  n  i1 e i  x  ⎛ ⎝ n  j1 C M ij  x  n j ⎞ ⎠ 2 , 2.25 thus the condition 2.13 is the form u|  M  0, M   ⎧ ⎨ ⎩ x ∈ ∂Ω | n  i,j1 a ij  x  n i n j > 0 ⎫ ⎬ ⎭ . 2.26 It shows that when m  1, 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 X   v ∈ C ∞  Ω  | D α v| ∂Ω  0, | α | ≤ m − 2, and v satisfy  2.13  ,  v  2 < ∞  , 2.27 where · 2 is defined by  v  2  ⎡ ⎣  Ω  | α | ≤m | D α v | 2 dx   ∂Ω  | γ | m−1 | D γ v | 2 ds ⎤ ⎦ 1/2 . 2.28 8 Boundary Value Problems We denote by X 2 the completion of X under the norm · 2 and by X 1 the completion of X with the following norm  v  1  ⎡ ⎢ ⎣  Ω ⎛ ⎝  | α |  | β | m a αβ  x  D α vD β v   | γ | ≤m−1 | D γ v | 2 ⎞ ⎠ dx   ∂Ω N m−1  i1 | h i  x  | ⎛ ⎝ N m−1  j1 C B ij D γ j v ⎞ ⎠ 2 ds ⎤ ⎥ ⎦ 1/2 . 2.29 Definition 2.4. u ∈ X 1 is a weak solution of 2.10–2.13 if for any v ∈ X 2 , the following equality holds:  Ω ⎡ ⎣  | α |  | β | m, | γ | m−1  a αβ  x  D β u  b αγ  x  D γ u  D α v   | θ | , | λ | ≤m−1 d θλ  x  D λ uD θ v ⎤ ⎦ dx − N m−1  i1   C i h i  x  ⎛ ⎝ N m−1  j1 C B ij D γ j u ⎞ ⎠ ⎛ ⎝ N m−1  j1 C B ij D γ j v ⎞ ⎠ ds   Ω f  x  vdx. 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 L are sufficiently smooth. Let u be a classical solution of 2.10–2.13. Denote by  Lu, v    Ω Lu · vdx, ∀v ∈ X. 2.31 Thanks to integration by part, we have  Ω Lu · vdx   Ω ⎡ ⎣  | α |  | β | m, | γ | m−1  a αβ  x  D β u  b αγ  x  D γ u  D α v   | θ | , | λ | ≤m−1 d θλ  x  D λ uD θ v ⎤ ⎦ dx −  ∂Ω ⎡ ⎣  | α |  | β | m a αβ  x  D β uD α−δ k v · n k   | λ |  | θ | m−1 n  i1 b i λθ  x  · n i D θ uD λ v ⎤ ⎦ ds. 2.32 Boundary Value Problems 9 Since v ∈ X, we have  ∂Ω  |α||β|m a αβ  x  D β uD α−δ k v · n k ds   ∂Ω N m  i1 e i  x  ⎛ ⎝ N m  j1 C M ij D α j u ⎞ ⎠ ⎛ ⎝ N m  j1 C M ij D α j −δ k j v · n k j ⎞ ⎠ ds  0. 2.33 Because u satisfies 2.12,  ∂Ω  |λ||θ|m−1 n  i1 b i λθ  x  · n i D θ uD λ vds   ∂Ω N m−1  i1 h i  x  ⎛ ⎝ N m−1  j1 C B ij D γ j u ⎞ ⎠ ⎛ ⎝ N m−1  j1 C B ij D γ j v ⎞ ⎠ ds  N m−1  i1   C i h i  x  ⎛ ⎝ N m−1  j1 C B ij D γ j u ⎞ ⎠ ⎛ ⎝ N m−1  j1 C B ij D γ j v ⎞ ⎠ ds. 2.34 From the three equalities above we obtain 2.30. Let u ∈ X 1 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 X 1 . In fact, we can show that if u ∈ X 1 , then u satisfies N m  i1   M i e i  x  ⎛ ⎝ N m  j1 C M ij D α j −δ k j u · n k j ⎞ ⎠ ⎛ ⎝ N m  j1 C M ij D α j v ⎞ ⎠ ds  0, ∀v ∈ X 1 ∩ W m1,2  Ω  . 2.35 Evidently, when u ∈ X, v ∈ X 1 ∩ W m1,2 Ω, we have  Ω  | α |  | β | m a αβ  x  D β uD α vdx −  Ω  | α |  | β | m D i  a αβ  x  D α v  D β−δ i udx. 2.36 If we can verify that for any u ∈ X 1 , 2.36 holds true, then we get  ∂Ω  |α||β|m a αβ  x  D α vD β−δ i u · n i ds  0, 2.37 10 Boundary Value Problems which means that 2.35 holds true. Since X is dense in X 1 ,foru ∈ X 1 given, let u k ∈ X and u k → u in X 1 . Then lim k →∞  Ω  |α||β|m a αβ D β u k D α vdx  Ω  |α||β|m a αβ D β uD α vdx, lim k →∞  Ω  |α||β|m D i  a αβ D α v  D β−δ i u k dx   Ω  |α||β|m D i  a αβ D α v  D β−δ i udx. 2.38 Due to u k satisfying 2.36, hence u ∈ X 1 satisfies 2.36.Thus2.31 is verified. Remark 2.5. When 2.2 is a diagonal matrix, then 2.13 is the form D γ u|  M γ  0, for   γ    m − 1, 2.39 where  M γ  {x ∈ ∂Ω |  n i1 a γδ iγ δ i x · n i 2 > 0}. 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 u of 2.10–2.13 belongs to X 1 ∩ W m,p Ω for some p>1, then by the trace embedding theorems, the condition 2.13 also holds true. It remains to verify the condition 2.12.Letu 0 ∈ X 1 ∩ W m1,2 Ω satisfy 2.30. Since W m1,2 Ω → X 2 , hence we have  Ω ⎡ ⎣  | α |  | β | m, | γ | m−1  a αβ  x  D β u 0  b αγ  x  D γ u 0  D α u 0   | θ | , | λ | ≤m−1 d θλ  x  D λ u 0 D θ u 0 − fu 0 ⎤ ⎦ ds − N m−1  i1   C i h i  x  ⎛ ⎝ N m−1  j1 C B ij D γ j u 0 ⎞ ⎠ 2 ds  0. 2.40 On the other hand, by 2.30, for any v ∈ C ∞ 0 Ω,weget  Ω ⎡ ⎣ −  | α |  | β | m D i  a αβ  x  D α u 0  D β−δ i v   | θ | , | λ | ≤m−1 d θλ  x  D λ u 0 D θ v −fv − D i ⎛ ⎝  | θ |  | γ | m−1 b i θγ  x  D γ u 0 ⎞ ⎠ D θ v ⎤ ⎦ dx  0. 2.41 [...]... then the problem 4.2 – 4.5 has a weak solution u ∈ W m,p Ω ∩ X1 , p 2δ/ 1 δ The proof of Theorem 4.4 is parallel to that of Theorem 4.3; here we omit the detail Acknowledgment This project was supported by the National Natural Science Foundation of China no 10971148 Boundary Value Problems 23 References 1 M V Keldys, “On certain cases of degeneration of equations of elliptic type on the boundry of. .. large enough 3.14 Ones can easily show that the mapping A : X1 → X2 ∗ is weakly continuous Here we omit the details of the proof By Lemma 2.7, this theorem is proven Boundary Value Problems 17 y ΣB ∩ ΣB 2 1 ΣB 2 θ x ΣB 1 Figure 2 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: ∂3 u ∂x3... Equation with Nonnegative Characteristic Form, Plennum Press, New York, NY, USA, 1973 3 T Ma and Q Y Yu, The Keldys-Fichera boundary value problems for degenerate quasilinear elliptic equations of second order,” Differential and Integral Equations, vol 2, no 4, pp 379–388, 1989 4 Z C Chen, The Keldys-Fichera boundary value problem for a class of nonlinear degenerate elliptic equations, ” Acta Mathematica... -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 , X2 be the 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 {un } ⊂... exists a sequence of {un } ⊂ X, un u0 in 0 Furthermore, if, we can derive that u Y < C, C is a constant, then the X1 such that Gun , un solution u0 of Gu 0 belongs to Y In the following, we give some existence theorems of W m,p -solutions for the boundary value conditions 4.3 – 4.5 of higher-order degenerate elliptic equations First, we consider the quasilinear equations −1 Au m Dα aαβ x, Du Dβ u bαγ x... 1 − C, ∀u ∈ X 2.50 Boundary Value Problems 13 Thus by Holder inequality see 13 , we have ¨ Lu − f, u ≥ 0, ∀u ∈ X, u 1 R great enough 2.51 By Lemma 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 Ω 1/p 0 in L, for all |α| m, |γ| m − 1, 1/q 1 , then such a solution... If all the odd terms bαγ x of L, then 2.30 holds for all v ∈ X1 , in the same fashion we known that the weak solution of 2.10 – 2.13 in X1 is unique The proof is complete Remark 2.10 In next subsection, we can see that under certain assumptions, the weak 1/q 1 solutions of degenerate elliptic equations are in X1 ∩ W m,p Ω ∩ W m−1,q Ω 1/p 3 Existence of Higher-Order Quasilinear Equations Given the quasilinear... 9, no 2, pp 203–211, 1993 5 Z.-C Chen and B.-J Xuan, “On the Keldys-Fichera boundary- value problem for degenerate quasilinear elliptic equations, ” Electronic Journal of Differential Equations, no 87, pp 1–13, 2002 6 S H Li, The first boundary value problem for quasilinear elliptic-parabolic equations with double degenerate,” Nonlinear Analysis: Theory, Methods & Applications, vol 27, no 1, pp 115–124,... subelliptic operators,” Journal of Functional Analysis, vol 199, no 1, ¨ pp 228–242, 2003 8 L M Li and T Ma, “Regularity of Keldys-Fichera boundary value problem for gegenerate elliptic equations, ” Chinese Annals of Mathematics, Series B, vol 31, no B 5–6 , pp 1–10, 2010 9 T Ma, Weakly continuous method and nonlinear differential equations with nonnegative characteristic form, Doctoral thesis, 1989 10 T Ma and... Figure 2 The odd term matrix is B x, y nx 0 0 ny x 0 0 y 3.16 It is easy to see that B {x ∈ ∂Ω | nx − x > 0} 1 π π 0 2 The boundary value condition associated with 3.15 is u|∂Ω ∂u ∂x ∂u ∂x 0, ∂u cos θ, sin θ ∂x B 1 B 2 ∂u cos θ, sin θ ∂x 0, 0, − π π . Corporation Boundary Value Problems Volume 2010, Article ID 208085, 23 pages doi:10.1155/2010/208085 Research Article The Boundary Value Problem of the Equations with Nonnegative Characteristic. of 3, 8–10. Boundary Value Problems 3 2. Formulation of the Boundary Value Problem For second-order equations with nonnegative characteristic form, Keldys 1 and Fichera presented a kind of. 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