2. Formulation of the Boundary Value Problem

Volume 2010, Article ID 208085,23pages 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

Copyrightq2010 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 and Young inequalities 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

Keldys1studies the boundary problem for linear elliptic equations with degenerationg on the boundary. For the linear elliptic equations with nonnegative characteristic forms, Oleinik and Radkevich2had discussed the Keldys-Fichera boundary value problem. In 1989, Ma and Yu3studied 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^mD^α

aαβxD^βubαγxD^γu

|θ|,|λ|≤m−1

−1^|θ|D^θ

d_θλxD^λu ,

1.1

wherex∈Ω, Ω⊂ Rⁿis an open set, the coefficients ofLare bounded measurable, and the leading term coefficients satisfy

aαβxξαξβ≥0. 1.2

We investigate the generalized Keldys-Fichera boundary value conditions as follows:

D^αu|_∂Ω0, |α| ≤m−2, 1.3

Nm−1

j1

C_ij^BxD^λju|^B_i 0, λ^jm−1, 1≤i≤N_m−1, 1.4

Nm

j1

C^M_ij xD^αj−δkju·n_kj|^M

i 0, ∀δkj ≤α^j, 1.5

with|α^j|mand 1≤i≤Nm, whereδkj {0, . . . , 1

kj

, . . . ,0}.

The leading term coefficients are symmetric, that is, aαβx aβαx which can be made into a symmetric matrixMx aαⁱα^j. The odd order term coefficientsb_θλxcan be made into a matrixBx _n

k1b_λⁱ_λ^j_x·nk,→−n n1, . . . , nnis the outward normal at∂Ω.

{eix}^N_i1^mand{hix}^N_i1^m−1are the eigenvalues of matricesMxandBx, respectively.C^B_ijx andC^M_ij xare orthogonal matrix satisfying

C_ij^MxMxC^M_ij x eixδij

i,j1,...,Nm, C^B_ijxBxC_ij^Bx

hixδij

i,j1,...,Nm−1.

1.6

The boundary sets are M

i

{x∈∂Ω|e_ix>0}, 1≤i≤N_m, B

i

{x∈∂Ω|h_ix>0}, 1≤i≤N_m−1.

1.7

At last, we study the existence and regularity of the following quasilinear differential operator with boundary conditions1.3–1.5:

Au

|α||^β|^m,|^γ|^m−1

−1^mD^α aαβ

x,

u

D^βubαγxD^γu

|^γ|^|θ|m−1

−1^m−1D^γ d_γθ

x, u

D^θu

|λ|≤m−1

−1^|λ|D^λg_λ x,

u ,

1.8

wherem≥2 and

u{D^αu}_|α|≤m−2.

This paper is a generalization of3,8–10.

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^mD^α

a_αβxD^βub_αγxD^γu

|θ|,|λ|≤m−1

−1^|θ|D^θ

dθλxD^λu ,

2.1

wherex∈Ω, Ω⊂Rⁿis an open set, the coefficients ofLare bounded measurable functions, andaαβx aβαx.

Let{gαβx}be a series of functions withg_αβ g_βα,|α||β|k. If in certain order we put all multiple indexesαwith|α| k into a row{α¹, . . . , α^Nk}, then{gαβx}can be made into a symmetric matrixg_αⁱ_α^j. By this rule, we get a symmetric leading term matrix of2.1, as follows:

Mx aαⁱα^jx_i,j1,...,Nm. 2.2

Suppose that the matrixMxis semipositive, that is,

0≤a_αiα^jxξiξ_j, ∀x∈Ω, ξ∈R^Nm, 2.3 and the odd order part of2.1can be written as

|α|m,|^γ|^m−1

−1^mD^α

b_αγxD^γu ⁿ

i1

|λ||θ|m−1

−1^mD^λδi

b_λθⁱ xD^θu

, 2.4

whereδi{δi1, . . . , δin}, δijis the Kronecker symbol. Assume that for all 1≤i≤n, we have

bⁱ_λθx bⁱ_θλx, x∈Ω. 2.5

We introduce another symmetric matrix

Bx _n

k1

b^k_λiλ^jx·n_k

i,j1,...,Nm−1

, x∈∂Ω, 2.6

where→−n {n1, n2, . . . , nn}is the outward normal atx ∈ ∂Ω. Let the following matrices be orthogonal:

C^Mx

C^M_ij x

i,j1,...,Nm

, x∈Ω, C^Bx

C_ij^Bx

i,j1,...,Nm−1, x∈∂Ω, 2.7 satisfying

C^MxMxC^Mx eixδij

i,j1,...,Nm, C^BxBxC^Bx

h_ixδij

i,j1,...,Nm−1,

2.8

where Cx is the transposed matrix ofCx,{eix}^N_i1^m are the eigenvalues of Mxand {hix}^N_i1^m−1are the eigenvalues ofBx. Denote by

M i

{x∈∂Ω|e_ix>0}, 1≤i≤N_m, B

i

{x∈∂Ω|hix>0}, 1≤i≤N_m−1, C

i

∂Ω\^B

i

, 1≤i≤N_m−1.

2.9

For multiple indicesα, β, α≤ βmeans thatαi ≤βi,for all 1≤i≤n. Now let us consider the following boundary value problem,

Lufx, x∈Ω, 2.10

D^αu|_∂Ω 0, |α| ≤m−2, 2.11

Nm−1

j1

C_ij^BxD^λju|^B

i 0, λ^jm−1, 1≤i≤N_m−1, 2.12

Nm

j1

C_ij^MxD^αj−δkju·nkj|^M_i 0, 2.13

for allδ_kj≤α^j, |α^j|mand 1≤i≤N_m, whereδ_kj{0, . . . , 1

kj

, . . . ,0}.

We can see that the item 2.13 of boundary value condition is determined by the leading term matrix 2.2, and 2.12is defined by the odd term matrix2.6. Moreover, if the operatorLis a not elliptic, then the operator

Lu

|θ|,|λ|≤m−1

−1^|θ|D^θ

dθλxD^λu

2.14

has to be elliptic.

In order to illustrate the boundary value conditions2.11–2.13, in the following we give an example.

Example 2.1. Given the differential equation

∂⁴u

∂x⁴₁ ∂⁴u

∂x₁²∂x₂² ∂³u

∂x³₂ −Δuf, x∈Ω⊂R². 2.15

HereΩ {x1, x₂∈R² |0< x₁ <1, 0< x₂ <1}. Letα¹{2,0}, α²{1,1}. α³{0,2}and λ¹{1,0}, λ²{0,1}, then the leading and odd term matrices of2.15respectively are

M

⎛

⎜⎜

⎝ 1 0 0 0 1 0 0 0 0

⎞

⎟⎟

⎠,

B 0 0

0 n₂

,

2.16

and the orthogonal matrices are

C^M

⎛

⎜⎜

⎝ 1 0 0 0 1 0 0 0 1

⎞

⎟⎟

⎠,

C^B 1 0

0 1

.

2.17

We can see that_M

1 ∂Ω, _M

2 ∂Ω, _M

3 φ, and_B

1 φ, _B

2 as shown inFigure 1.

The item2.12is 2 j1

C^B_2jD^λju|^B

2 D^λ2u|^B

2 ∂u

∂x_{2 B}

2

0, 2.18

x2

x₁ Σ^B2

Γ Ω

Figure 1

and the item2.13is 3 j1

C^M_1jD^αj−δkju·n_kj|^M

1 D^α1−δk1u·n_k1|^M

1 0,

3 j1

C^M_2jD^αj−δkju·n_kj|^M

2 D^α2−δk2u·n_k2|^M

2 0,

2.19

for allδ_k1≤α¹andδ_k2≤α². Since onlyδ_k1{1,0} ≤α¹{2,0}, hence we have D^α1−δk1u·n_k1|^M

1 ∂u

∂x₁ ·n₁|_∂Ω 0, 2.20

however,δk2 {1,0}< α² {1,1}andδk2{0,1}< α², therefore,

D^α2−δk2u·nk2|^M

2

⎧⎪

⎪⎨

⎪⎪

⎩

∂u

∂x2 ·n1|_∂Ω 0,

∂u

∂x₁ ·n₂|_∂Ω 0.

2.21

Thus the associated boundary value condition of2.15is as follows:

u|_∂Ω 0, ∂u

∂x2

∂Ω/Γ0, ∂u

∂x1

∂Ω0, 2.22

which implies that∂u/∂x2is free onΓ {x1, x2∈∂Ω|0< x1<1, x20}.

Remark 2.2. In general the matrices Mx and Bx arranged are not unique, hence the boundary value conditions relating to the operatorLmay not be unique.

Remark 2.3. When all leading terms ofLare zero,2.10is an odd order one. In this case, only 2.11and2.12remain.

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 problem2.10–2.13is the Dirichlet problem provided the operatorLbeing ellipticsee11. In this case,_M

i ∂Ωfor all 1≤i≤Nm. Besides,2.13 run over all 1 ≤ i ≤ Nm and δkj ≤ αⁱ, moreoverC^Bxis nondegenerate for anyx ∈ ∂Ω.

Solving the system of equations, we getD^αu|_∂Ω 0, for all |α|m−1.

Whenm1, namely,Lis of second-order, the condition2.12is the form

u|^B 0, B

x∈∂Ω|ⁿ

i1

b_ixni>0

, 2.23

and2.13is

n j1

C^M_ij xnju|^M

i 0, 1≤i≤n. 2.24

Noticing

n i,j1

a_ijxnin_j ⁿ

i1

e_ix

⎛

⎝ⁿ

j1

C^M_ij xnj

⎞

⎠

2

, 2.25

thus the condition2.13is the form

u|^M 0, M

⎧⎨

⎩x∈∂Ω|ⁿ

i,j1

a_ijxnin_j>0

⎫⎬

⎭. 2.26

It shows that whenm1,2.12and2.13are coincide with Keldys-Fichera boundary value condition.

Next, we will give the definition of weak solutions of2.10–2.13 see12. Let

X v∈C^∞ Ω

|D^αv|_∂Ω0, |α| ≤m−2, andv satisfy2.13, v₂<∞!

, 2.27

where · ₂is defined by

v₂

⎡

⎣$

Ω

|α|≤m

|D^αv|²dx

$

∂Ω

|^γ|^m−1

|D^γv|²ds

⎤

⎦

1/2

. 2.28

We denote byX2the completion ofX under the norm · ₂and by X1 the completion ofX with the following norm

v₁

⎡

⎢⎣

$

Ω

⎛

⎝

|α||^β|^m

a_αβxD^αvD^βv

|^γ|^≤m−1

|D^γv|²

⎞

⎠dx

$

∂Ω N_m−1

i1

|hix|

⎛

⎝^Nm−1

j1

C^B_ijD^γjv

⎞

⎠

2

ds

⎤

⎥⎦

1/2

.

2.29

Definition 2.4. u ∈ X1 is a weak solution of 2.10–2.13if for any v ∈ X2, the following equality holds:

$

Ω

⎡

⎣

|α||^β|^m,|^γ|^m−1

aαβxD^βubαγxD^γu

D^αv

|θ|,|λ|≤m−1

dθλxD^λuD^θv

⎤

⎦dx

−^Nm−1

i1

$

_C

i

hix

⎛

⎝^Nm−1

j1

C^B_ijD^γju

⎞

⎠

⎛

⎝^Nm−1

j1

C_ij^BD^γjv

⎞

⎠ds

$

Ωfxv dx.

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 ofLare sufficiently smooth.

Letube a classical solution of2.10–2.13. Denote by

Lu, v

$

ΩLu·v dx, ∀v∈X. 2.31

Thanks to integration by part, we have

$

ΩLu·v dx

$

Ω

⎡

⎣

|α||^β|^m,|^γ|^m−1

aαβxD^βubαγxD^γu

D^αv

|θ|,|λ|≤m−1

dθλxD^λuD^θv

⎤

⎦dx

−

$

∂Ω

⎡

⎣

|α||^β|^m

aαβxD^βuD^α−δkv·nk

|λ||θ|m−1

n i1

bⁱ_λθx·niD^θuD^λv

⎤

⎦ds.

2.32

Sincev∈X, we have

$

∂Ω

|α||β|m

a_αβxD^βuD^α−δkv·n_kds

$

∂Ω Nm

i1

e_ix

⎛

⎝^Nm

j1

C_ij^MD^αju

⎞

⎠

⎛

⎝^Nm

j1

C^M_ij D^αj−δkjv·n_kj

⎞

⎠ds0.

2.33

Becauseusatisfies2.12,

$

∂Ω

|λ||θ|m−1

n i1

bⁱ_λθx·niD^θuD^λv ds

$

∂Ω Nm−1

i1

hix

⎛

⎝^Nm−1

j1

C^B_ijD^γju

⎞

⎠

⎛

⎝^Nm−1

j1

C_ij^BD^γjv

⎞

⎠ds

^Nm−1

i1

$

C i

hix

⎛

⎝^Nm−1

j1

C^B_ijD^γju

⎞

⎠

⎛

⎝^Nm−1

j1

C^B_ijD^γjv

⎞

⎠ds.

2.34

From the three equalities above we obtain2.30.

Letu ∈ X₁ be a weak solution of2.10–2.13. Then the boundary value conditions 2.11and2.13can be reflected by the spaceX1. In fact, we can show that ifu∈X1, thenu satisfies

Nm

i1

$

M i

eix

⎛

⎝^Nm

j1

C^M_ij D^αj−δkju·nkj

⎞

⎠

⎛

⎝^Nm

j1

C^M_ij D^αjv

⎞

⎠ds0, ∀v∈X1∩W^m1,2Ω.

2.35

Evidently, whenu∈X, v∈X₁∩W^m1,2Ω, we have

$

Ω

|α||^β|^m

aαβxD^βuD^αv dx−

$

Ω

|α||^β|^m Di

aαβxD^αv

D^β−δiu dx. 2.36

If we can verify that for anyu∈X₁,2.36holds true, then we get

$

∂Ω

|α||β|m

aαβxD^αvD^β−δiu·nids0, 2.37

which means that2.35holds true. SinceXis dense inX1, foru∈X1given, letuk ∈Xand u_k → uinX₁. Then

klim→ ∞

$

Ω

|α||β|m

aαβD^βukD^αv dx

$

Ω

|α||β|m

aαβD^βuD^αv dx,

klim→ ∞

$

Ω

|α||β|m

D_i

a_αβD^αv

D^β−δiu_kdx

$

Ω

|α||β|m

D_i

a_αβD^αv

D^β−δiu dx.

2.38

Due tou_ksatisfying2.36, henceu∈X₁satisfies2.36. Thus2.31is verified.

Remark 2.5. When2.2is a diagonal matrix, then2.13is the form

D^γu|^M

γ 0, forγm−1, 2.39

where _M

γ {x ∈ ∂Ω | _n

i1a_γδiγδix·ni2 > 0}. In this case, the corresponding trace embedding theorem can be set, and the boundary value condition2.13is naturally satisfied.

On the other hand, if the weak solutionuof2.10–2.13belongs toX1∩W^m,pΩfor some p >1, then by the trace embedding theorems, the condition2.13also holds true.

It remains to verify the condition2.12. Letu₀ ∈X₁∩W^m1,2Ωsatisfy2.30. Since W^m1,2Ω→X2, hence we have

$

Ω

⎡

⎣

|α||^β|^m,|^γ|^m−1

aαβxD^βu0bαγxD^γu0

D^αu0

|θ|,|λ|≤m−1

dθλxD^λu0D^θu0−fu0

⎤

⎦ds

−^Nm−1

i1

$

C i

h_ix

⎛

⎝^Nm−1

j1

C^B_ijD^γju₀

⎞

⎠

2

ds0.

2.40

On the other hand, by2.30, for anyv∈C^∞₀ Ω, we get

$

Ω

⎡

⎣−

|α||^β|^m Di

aαβxD^αu0

D^β−δiv

|θ|,|λ|≤m−1

dθλxD^λu0D^θv

−fv−D_i

⎛

⎝

|θ||^γ|^m−1

b_θγⁱ xD^γu₀

⎞

⎠D^θv

⎤

⎦dx0.

2.41

Because the coefficients ofLare sufficiently smooth, andC₀^∞is dense inW₀^m−1,2Ω, equality 2.41also holds for anyv∈W₀^m−1,2Ω. Therefore, due tou0∈W₀^m−1,2Ω, we have

$

Ω

⎡

⎣−

|α||^β|^m D_i

a_αβxD^αu₀

D^β−δiu₀

|θ|,|λ|≤m−1

d_θλxD^λu₀D^θu₀

−fu0−D_i

⎛

⎝

|θ||γ|m−1

bⁱ_θγxD^γu₀

⎞

⎠D^θu₀

⎤

⎦dx0.

2.42

From2.36, one drives

−

$

Ω

|α||β|m

Di

aαβxD^αu0

D^β−δiu0dx

$

Ω

|α||β|m

aαβxD^αu0D^βu0dx, 2.43

Furthermore,

−

$

ΩDi

⎛

⎝

|θ||^γ|^m−1

bⁱ_θγxD^γu0

⎞

⎠D^θu0dx

$

Ω

|α|m,|^γ|^m−1

bαγxD^γu0D^αu0dx−^Nm−1

i1

$

_C

i ∪_B

i

hix

⎛

⎝^Nm−1

j1

C^B_ijD^γju0

⎞

⎠

2

ds.

2.44

From2.30and2.42, one can see that

Nm−1

i1

$

B i

hix

⎛

⎝^Nm−1

j1

C^B_ijD^γju0

⎞

⎠

2

ds0. 2.45

Noticingh_ix>0 in_B

i, one deduces thatu₀satisfies2.12providedu₀ ∈X₁∩W^m1,2Ω.

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

LetXbe a linear space, andX₁, X₂be the completion ofX, respectively, with the norm · ₁, · ₂. Suppose thatX₁is a reflexive Banach space andX₂is a separable Banach space.

Definition 2.6. A mappingG:X1 → X2∗is called to be weakly continuous, if for anyxn, x0∈ X₁, x_n x₀inX₁, one has

nlim→ ∞

)Gxn, y* )

Gx0, y*

, ∀y∈X2. 2.46

Lemma 2.7see3. Suppose thatG:X1 → X2∗is a weakly continuous, if there exists a bounded open setΩ⊂X₁, such that

Gu, u ≥0, ∀u∈∂Ω∩X, 2.47

then the equationGu0 has a solution inX₁.

Theorem 2.8existence theorem. LetΩ ⊂ Rⁿ be an arbitrary open set, f ∈ L²Ω andbαγ ∈ C¹Ω. If there exist a constantC >0 andg∈L¹Ωsuch that

C

|γ|m−1

ξ_γ²C|ξi|²−g≤

|λ|,|θ|≤m−1

d_θλxξθξ_λ−1 2

n i1

|γ||β|m−1

D_ib_γβⁱ xξγξ_β, 2.48

whereξαis the component of ξ∈R^Nm−1corresponding toD^αu, then the problem2.10–2.13has a weak solution inX₁.

Proof. LetLu, vbe the inner product as in2.31. It is easy to verify thatLu, vdefines a bounded linear operatorL:X1 → X2∗. HenceLis weakly continuoussee3. From2.42, foru∈Xwe drive that

Lu, u

$

Ω

⎡

⎣

|α||^β|^m

aαβxD^αuD^βuⁿ

i1

|λ||θ|m−1

b_λθⁱ xD^θuD^λδiu

|γ|,|α|≤m−1

d_γαxD^γuD^αu

⎤

⎦dx

−^Nm−1

i1

$

C i

h_ix

⎛

⎝^Nm−1

j1

C^B_ijD^γju

⎞

⎠

2

ds

$

Ω

⎡

⎣

|α||^β|^m

a_αβxD^αuD^βu

|^γ|^{,|α|≤m−1}

d_γαxD^γuD^αu

−1 2

n i1

|γ||β|m−1

D_ibⁱ_γβxD^γuD^βu

⎤

⎦dx

1 2

Nm−1

i1

⎡

⎢⎣

$

B i

−

$

C i

hix

⎛

⎝^Nm−1

j1

C^B_ijD^γju

⎞

⎠

2⎤

⎥⎦ds

≥

$

Ω

⎡

⎣

|α||^β|^m

aαβxD^αuD^βuC

|^γ|^m−1|D^γu|²Cu²−gx

⎤

⎦

1 2

Nm−1

i1

⎡

⎢⎣

$

B i∪C

i

|hix|

⎛

⎝^Nm−1

j1

C^B_ijxD^γju

⎞

⎠

2⎤

⎥⎦ds.

2.49

Hence we obtain

Lu, u ≥Cu²₁−C, ∀u∈X. 2.50

Thus by H ¨older inequalitysee13, we have )Lu−f, u*

≥0, ∀u∈X, u₁Rgreat enough. 2.51

ByLemma 2.7, the theorem is proven.

Theorem 2.9 uniqueness theorem. Under the assumptions of Theorem 2.8 with gx 0 in 2.48. If the problem 2.10–2.13 has a weak solution inX1 ∩W^m,pΩ∩W^m−1,qΩ1/p 1/q 1, then such a solution is unique. Moreover, ifbαγx 0 inL, for all|α|m, |γ|m−1, then the weak solutionu∈X₁of2.10–2.13is unique.

Proof. Letu0 ∈X1∩W^m,pΩ∩W^m−1,q be a weak solution of2.10–2.13. We can see that 2.30holds for allv∈X1∩W^m,p∩W^m−1,qΩ. HenceLu0, u0is well defined. Letu1 ∈X1∩ W^m,p∩W^m−1,qΩ. Then from2.49it follows that< Lu₁−Lu₀, u₁−u₀ >0, we obtainu₁u₀, which means that the solution of2.10–2.13inX1∩W^m,p∩W^m−1,qΩis unique. If all the odd termsbαγxofL, then2.30holds for allv∈X1, in the same fashion we known that the weak solution of2.10–2.13inX₁is 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 inX1∩W^m,pΩ∩W^m−1,qΩ1/p1/q 1.

3. Existence of Higher-Order Quasilinear Equations

Given the quasilinear differential operator

Au

|α||^β|^m,|^γ|^m−1

−1^mD^α aαβ

x,

u

D^βubαγxD^γu

|^γ|^|θ|m−1

−1^m−1D^γ d_γθ

x, u

D^θu

|λ|≤m−1

−1^|λ|D^λg_λ x,

u ,

3.1

wherem≥2 and

u{D^αu}_|α|≤m−2.

Leta_αβx, ξ a_βαx, ξ, the odd order part of3.1be as that in2.4,b_αγ∈C¹Ω, and _B

i

_C

i,be the same as those inSection 2. The leading matrix is

Mx, ξ aαⁱα^jx, ξ_i,j1,...,Nm, 3.2

and the eigenvalues are{eix, ξ}^N_i1^m. We denote_M

i {x∈∂Ω|e_ix,0>0}, 1≤i≤N_m.

We consider the following problem:

Aufx, x∈Ω, u|_∂Ω 0,

Nm−1

j1

C^B_ijxD^λju|^B_i 0, λ^jm−1, 1≤i≤N_m−1,

Nm

j1C^M_ij x,0D^αj−δkju·nkj|^M

i 0, ∀δkj ≤α^j, with α^jm, 1≤i≤N_m, δ_kj

⎧⎪

⎨

⎪⎩0, . . . , 1

kj

, . . . ,0

⎫⎪

⎬

⎪⎭.

3.3

Denote the anisotropic Sobolev space by W_|α|≤k^pα Ω +

u∈L^p0Ω|p0≥1, D^αu∈L^pαΩ, ∀1≤ |α| ≤k, and pα≥1, orpα0, , 3.4

whose norm is

u

|α|≤k

signpαD^αu_Lpα, 3.5

when allp_α pfor|α| k, then the space is denoted byW_{k,|α|≤k−1}^p,pα Ω.q_θ|θ| ≤kis termed the critical embedding exponent fromW_k,|α|≤k^pα ΩtoL^pΩ, ifqθis the largest number of the exponentpin whereD^θu∈L^pΩ, for allu∈W_|α|≤k^pα Ω, and the embedding is continuous.

For example, whenΩis bounded, the spaceX {u∈L^kΩ|k ≥1, Diu∈L²Ω,1 ≤ i ≤ n} with norm u ∇u_L² u_L^k is an anisotropic Sobolev space, and the critical embedding exponents fromXtoL^PΩareqi21≤i≤n, andq0max{k,2n/n−2}.

Suppose that the following hold.

A1The coefficients of the leading term ofAsatisfy one of the following two conditions:

1a_αβx, η a_αβx;

2aαβx, η 0, asα /β.

A2There is a constantM >0 such that

0≤M

|α||β|m

aαβx,0ξαξβ≤

|α||β|m

aαβ

x, η ξαξβ

≤M⁻¹

|α||β|m

aαβx,0ξαξβ.

3.6

A3There are functionsGix, η i0,1, . . . , nwithGix,0 0, for all 1≤i≤n, such that

|γ|m−1

gγ

x,

u

Dγuⁿ

i1

DiGi

x,

u G0

x,

u

. 3.7

A4There is a constantC >0 such that

C|ξ|²≤

|α||^β|^m−1 -

d_αβxξαξ_β− 1 2

n i1

D_ibⁱ_αβxξαξ_β .

,

C

|λ|≤m−1

signp_λη_λ^pλ−f₁≤

|θ|≤m−2

g_θ x, η

η_θG₀ x, η

,

3.8

wheref1∈L¹Ω, p0>1, pλ>1 orpλ0, for all 1≤ |λ| ≤m−2.

A5There is a constantc >0 such that aαβ

x, η≤C, dγθ

x, η≤C

⎡

⎣

|^β|^≤m−2

ηβ^Sβ1

⎤

⎦,

g_γ

x, η≤C

⎡

⎣

|^β|^≤m−2

η_β^Sβ1

⎤

⎦,

3.9

where 1 ≤ Sβ < qβ/2, 1 ≤ Sβ < qβ, qβ is a critical embedding exponent from Wm−1,|λ|≤m−1^2,pλ ΩtoL^PΩ. LetXbe defined by2.27andX₁be the completion ofX under the norm

v₁

⎡

⎣$

Ω

⎛

⎝

|α||^β|^m

aαβx,0D^αvD^βv

|^γ|^m−1|D^γv|²

⎞

⎠dx

$

∂Ω N_m−1

i1

|hix|

⎛

⎝^Nm−1

j1

C^B_ijD^γjv

⎞

⎠

2

ds

⎤

⎥⎦

1/2

|γ|≤m−2

signp_γD^γv_L^pγ,

3.10