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,[email protected] Tian Ma,[email protected]
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 inequaliinequali-ties 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
−1mDαaαβxDβubαγxDγu
|θ|,|λ|≤m−1
−1|θ|DθdθλxDλu
,
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:
Dαu|∂Ω0, |α| ≤m−2, 1.3
Nm−1
j1
CijBxDλju|B i 0,
λjm−1, 1≤i≤Nm−1, 1.4
Nm
j1
CMij xDαj−δkju·n kj|M
i 0, ∀δkj ≤α
j, 1.5
with|αj|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 aαiαj. The odd order term coefficientsbθλxcan be
made into a matrixBx nk1bλiλjx·nk,→−n n1, . . . , nnis the outward normal at∂Ω.
{eix}Nmi1 and{hix}iNm1−1are the eigenvalues of matricesMxandBx, respectively.CBijx
andCMij xare orthogonal matrix satisfying
CijMxMxCMij xeixδij
i,j1,...,Nm,
CBijxBxCijBxhixδij
i,j1,...,Nm−1.
1.6
The boundary sets are
M
i
{x∈∂Ω|eix>0}, 1≤i≤Nm,
B
i
{x∈∂Ω|hix>0}, 1≤i≤Nm−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
−1mDαaαβ
x,uDβubαγxDγu
|γ||θ|m−1
−1m−1Dγdγθ
x,uDθu
|λ|≤m−1
−1|λ|Dλgλ
x,u,
1.8
wherem≥2 andu{Dαu}
|α|≤m−2.
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
−1mDαa
αβxDβubαγxDγu
|θ|,|λ|≤m−1
−1|θ|DθdθλxDλu
,
2.1
wherex∈Ω, Ω⊂Rnis an open set, the coefficients ofLare bounded measurable functions,
andaαβx aβαx.
Let{gαβx}be a series of functions withgαβ gβα,|α||β|k. If in certain order we
put all multiple indexesαwith|α| k into a row{α1, . . . , αNk}, then{g
αβx}can be made
into a symmetric matrixgαiαj. By this rule, we get a symmetric leading term matrix of2.1,
as follows:
Mx aαiαjxi,j1,...,Nm. 2.2
Suppose that the matrixMxis semipositive, that is,
0≤aαiαjxξiξj, ∀x∈Ω, ξ∈RNm, 2.3
and the odd order part of2.1can be written as
|α|m,|γ|m−1
−1mDαbαγxDγu
n
i1
|λ||θ|m−1
−1mDλδibλθi xDθu, 2.4
whereδi{δi1, . . . , δin}, δijis the Kronecker symbol. Assume that for all 1≤i≤n, we have
bi
λθx biθλx, x∈Ω. 2.5
We introduce another symmetric matrix
Bx
n
k1
bkλiλjx·nk
i,j1,...,Nm−1
where→−n {n1, n2, . . . , nn}is the outward normal atx ∈ ∂Ω. Let the following matrices be
orthogonal:
CMx CMij x
i,j1,...,Nm, x∈Ω,
CBx CijBx
i,j1,...,Nm−1
, x∈∂Ω,
2.7
satisfying
CMxMxCMxe ixδij
i,j1,...,Nm,
CBxBxCBxhixδij
i,j1,...,Nm−1,
2.8
where Cx is the transposed matrix ofCx,{eix}Nmi1 are the eigenvalues of Mxand
{hix}iNm1−1are the eigenvalues ofBx. Denote by
M
i
{x∈∂Ω|eix>0}, 1≤i≤Nm,
B
i
{x∈∂Ω|hix>0}, 1≤i≤Nm−1,
C
i
∂Ω\
B
i
, 1≤i≤Nm−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
Dαu|∂Ω 0, |α| ≤m−2, 2.11
Nm−1
j1
CijBxDλju|B i 0,
λjm−1, 1≤i≤Nm−1, 2.12
Nm
j1
CijMxDαj−δkju·n kj|M
i 0, 2.13
for allδkj≤αj, |αj|mand 1≤i≤Nm, whereδkj{0 , . . . ,1
kj
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|DθdθλxDλu
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
∂x4 1
∂4u
∂x2 1∂x22
∂3u
∂x3 2
−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
CM
⎛ ⎜ ⎜ ⎝
1 0 0
0 1 0
0 0 1
⎞ ⎟ ⎟ ⎠,
CB
1 0
0 1
.
2.17
We can see thatM1 ∂Ω, M2 ∂Ω, M3 φ, andB1 φ, B2 as shown inFigure 1. The item2.12is
2
j1
CB2jDλju|B
2 D
λ2
u|B
2
∂u ∂x2
B
2
x2
x1
ΣB
2
Γ Ω
Figure1
and the item2.13is
3
j1
CM1jDαj−δkju·n kj|M
1 D
α1−δk
1u·nk1|M
1 0,
3
j1
CM2jDαj−δkju·n kj|M
2 D
α2−δk 2u·nk
2|M2 0,
2.19
for allδk1≤α1andδk2≤α2. Since onlyδk1{1,0} ≤α1{2,0}, hence we have
Dα1−δk1u·nk 1|M1
∂u ∂x1 ·
n1|∂Ω 0, 2.20
however,δk2 {1,0}< α2 {1,1}andδk2{0,1}< α2, therefore,
Dα2−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
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,Mi ∂Ωfor all 1≤i≤Nm. Besides,2.13
run over all 1 ≤ i ≤ Nm and δkj ≤ αi, moreoverCBxis nondegenerate for anyx ∈ ∂Ω.
Solving the system of equations, we getDαu|
∂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
bixni>0
, 2.23
and2.13is
n
j1
CMij xnju|M
i 0, 1≤i≤n. 2.24
Noticing
n
i,j1
aijxninj n
i1
eix
⎛ ⎝n
j1
CMij xnj
⎞ ⎠
2
, 2.25
thus the condition2.13is the form
u|M 0, M
⎧ ⎨
⎩x∈∂Ω|
n
i,j1
aijxninj>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|Dαv|
∂Ω0, |α| ≤m−2, andv satisfy2.13, v2<∞
!
, 2.27
where · 2is defined by
v2
⎡ ⎣$
Ω
|α|≤m
|Dαv|2dx
$
∂Ω
|γ|m−1
|Dγv|2ds
⎤ ⎦
1/2
We denote byX2the completion ofX under the norm · 2and by X1 the completion ofX with the following norm
v1
⎡ ⎢ ⎣ $
Ω
⎛ ⎝
|α||β|m
aαβxDαvDβv
|γ|≤m−1
|Dγv|2
⎞ ⎠dx
$
∂Ω
Nm−1
i1
|hix|
⎛ ⎝Nm−1
j1
CBijDγjv
⎞ ⎠
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
aαβxDβubαγxDγu
Dαv
|θ|,|λ|≤m−1
dθλxDλuDθv
⎤ ⎦dx
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CBijDγju
⎞ ⎠
⎛ ⎝Nm−1
j1
CijBDγjv
⎞ ⎠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
aαβxDβubαγxDγu
Dαv
|θ|,|λ|≤m−1
dθλxDλuDθv
⎤ ⎦dx
− $
∂Ω
⎡ ⎣
|α||β|m
aαβxDβuDα−δkv·nk
|λ||θ|m−1
n
i1
biλθx·niDθuDλv
⎤ ⎦ds.
Sincev∈X, we have
$
∂Ω
|α||β|m
aαβxDβuDα−δkv·nkds
$
∂Ω
Nm
i1
eix
⎛ ⎝Nm
j1
CijMDαju
⎞ ⎠
⎛ ⎝Nm
j1
CMij Dαj−δkjv·n kj
⎞ ⎠ds0.
2.33
Becauseusatisfies2.12,
$
∂Ω
|λ||θ|m−1
n
i1
biλθx·niDθuDλv ds
$
∂Ω
Nm−1
i1
hix
⎛ ⎝Nm−1
j1
CBijDγju
⎞ ⎠
⎛ ⎝Nm−1
j1
CijBDγjv
⎞ ⎠ds
Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CBijDγju
⎞ ⎠
⎛ ⎝Nm−1
j1
CBijDγjv
⎞ ⎠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
CMij Dαj−δkju·n kj
⎞ ⎠
⎛ ⎝Nm
j1
CMij Dαjv
⎞
⎠ds0, ∀v∈X1∩Wm1,2Ω.
2.35
Evidently, whenu∈X, v∈X1∩Wm1,2Ω, we have
$
Ω
|α||β|m
aαβxDβuDαv dx−
$
Ω
|α||β|m
Di
aαβxDαv
Dβ−δiu dx. 2.36
If we can verify that for anyu∈X1,2.36holds true, then we get
$
∂Ω
|α||β|m
which means that2.35holds true. SinceXis dense inX1, foru∈X1given, letuk ∈Xand
uk → uinX1. Then
lim
k→ ∞
$
Ω
|α||β|m
aαβDβukDαv dx
$
Ω
|α||β|m
aαβDβuDαv dx,
lim
k→ ∞
$
Ω
|α||β|m
Di
aαβDαv
Dβ−δiukdx
$
Ω
|α||β|m
Di
aαβDαv
Dβ−δiu dx.
2.38
Due touksatisfying2.36, henceu∈X1satisfies2.36. Thus2.31is verified.
Remark 2.5. When2.2is a diagonal matrix, then2.13is the form
Dγu|M
γ 0, forγm−1, 2.39
where Mγ {x ∈ ∂Ω | ni1aγδiγδix·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∩Wm,pΩfor some
p >1, then by the trace embedding theorems, the condition2.13also holds true.
It remains to verify the condition2.12. Letu0 ∈X1∩Wm1,2Ωsatisfy2.30. Since
Wm1,2Ω→X2, hence we have
$
Ω
⎡
⎣
|α||β|m,|γ|m−1
aαβxDβu0bαγxDγu0
Dαu0
|θ|,|λ|≤m−1
dθλxDλu0Dθu0−fu0
⎤ ⎦ds
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CBijDγju0
⎞ ⎠
2
ds0.
2.40
On the other hand, by2.30, for anyv∈C∞0 Ω, we get
$
Ω
⎡ ⎣−
|α||β|m
Di
aαβxDαu0
Dβ−δiv
|θ|,|λ|≤m−1
dθλxDλu0Dθv
−fv−Di
⎛ ⎝
|θ||γ|m−1
bθγi xDγu0
⎞ ⎠Dθv
⎤ ⎦dx0.
Because the coefficients ofLare sufficiently smooth, andC0∞is dense inW0m−1,2Ω, equality
2.41also holds for anyv∈W0m−1,2Ω. Therefore, due tou0∈W0m−1,2Ω, we have
$
Ω
⎡ ⎣−
|α||β|m
Di
aαβxDαu0
Dβ−δiu0
|θ|,|λ|≤m−1
dθλxDλu0Dθu0
−fu0−Di
⎛ ⎝
|θ||γ|m−1
biθγxDγu0
⎞ ⎠Dθu
0
⎤ ⎦dx0.
2.42
From2.36, one drives
− $
Ω
|α||β|m
Di
aαβxDαu0
Dβ−δiu0dx
$
Ω
|α||β|m
aαβxDαu0Dβu0dx, 2.43
Furthermore,
− $
ΩDi
⎛ ⎝
|θ||γ|m−1
biθγxDγu0
⎞ ⎠Dθu
0dx
$
Ω
|α|m,|γ|m−1
bαγxDγu0Dαu0dx−
Nm−1
i1
$
C i ∪
B i
hix
⎛ ⎝Nm−1
j1
CBijDγju0
⎞ ⎠
2
ds.
2.44
From2.30and2.42, one can see that
Nm−1
i1
$
B i
hix
⎛ ⎝Nm−1
j1
CBijDγju0
⎞ ⎠
2
ds0. 2.45
Noticinghix>0 in
B
i, one deduces thatu0satisfies2.12providedu0 ∈X1∩Wm1,2Ω. 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, xn x0inX1, one has
lim
n→ ∞
)
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
Theorem 2.8existence theorem. LetΩ ⊂ Rn be an arbitrary open set, f ∈ L2Ω andb
αγ ∈
C1Ω. If there exist a constantC >0andg∈L1Ωsuch that
C
|γ|m−1
ξγ2C|ξi|2−g≤
|λ|,|θ|≤m−1
dθλxξθξλ−
1 2
n
i1
|γ||β|m−1
Dibγβi xξγξβ, 2.48
whereξαis the component of ξ∈RNm−1corresponding toDαu, 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
aαβxDαuDβu n
i1
|λ||θ|m−1
bλθi xDθuDλδiu
|γ|,|α|≤m−1
dγαxDγuDαu
⎤ ⎦dx
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CBijDγju
⎞ ⎠ 2 ds $ Ω ⎡ ⎣
|α||β|m
aαβxDαuDβu
|γ|,|α|≤m−1
dγαxDγuDαu
−1
2
n
i1
|γ||β|m−1
DibiγβxDγuDβu
⎤ ⎦dx
1
2
Nm−1
i1
⎡ ⎢ ⎣ $ B i − $ C i
hix
⎛ ⎝Nm−1
j1
CBijDγju
⎞ ⎠ 2⎤ ⎥ ⎦ds ≥ $ Ω ⎡ ⎣
|α||β|m
aαβxDαuDβuC
|γ|m−1
|Dγu|2Cu2−gx
⎤ ⎦
1
2
Nm−1
i1
⎡ ⎢ ⎣ $ B i∪ C i
|hix|
⎛ ⎝Nm−1
j1
CB ijxDγ
j u ⎞ ⎠ 2⎤ ⎥ ⎦ds. 2.49
Hence we obtain
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 ofTheorem 2.8 with gx 0 in 2.48. If the problem 2.10–2.13 has a weak solution inX1 ∩Wm,pΩ∩Wm−1,qΩ1/p
1/q 1, then such a solution is unique. Moreover, ifbαγx 0inL, for all|α|m, |γ|m−1,
then the weak solutionu∈X1of2.10–2.13is unique.
Proof. Letu0 ∈X1∩Wm,pΩ∩Wm−1,q be a weak solution of2.10–2.13. We can see that
2.30holds for allv∈X1∩Wm,p∩Wm−1,qΩ. HenceLu0, u0is well defined. Letu1 ∈X1∩
Wm,p∩Wm−1,qΩ. Then from2.49it follows that< Lu
1−Lu0, u1−u0 >0, we obtainu1u0, which means that the solution of2.10–2.13inX1∩Wm,p∩Wm−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∩Wm,pΩ∩Wm−1,qΩ1/p1/q 1.
3. Existence of Higher-Order Quasilinear Equations
Given the quasilinear differential operator
Au
|α||β|m,|γ|m−1
−1mDαaαβ
x,uDβubαγxDγu
|γ||θ|m−1
−1m−1Dγdγθ
x,uDθu
|λ|≤m−1
−1|λ|Dλg λ
x,u,
3.1
wherem≥2 andu{Dαu}
|α|≤m−2.
Letaαβx, ξ aβαx, ξ, the odd order part of3.1be as that in2.4,bαγ∈C1Ω, and
B
i
C
i,be the same as those inSection 2. The leading matrix is
Mx, ξ aαiαjx, ξi,j1,...,Nm, 3.2
We consider the following problem:
Aufx, x∈Ω,
u|∂Ω 0,
Nm−1
j1
CBijxDλju|B i 0,
λjm−1, 1≤i≤Nm−1,
Nm
j1
CMij x,0Dαj−δkju·n kj|M
i 0, ∀δkj ≤α j,
with αjm, 1≤i≤N
m, δkj
⎧ ⎪ ⎨ ⎪ ⎩0 , . . . ,1
kj
, . . . ,0
⎫ ⎪ ⎬ ⎪ ⎭.
3.3
Denote the anisotropic Sobolev space by
W|pαα|≤kΩ +u∈Lp0Ω|p
0≥1, Dαu∈LpαΩ, ∀1≤ |α| ≤k, and pα≥1, orpα0
,
,
3.4
whose norm is
u
|α|≤k
signpαDαuLpα, 3.5
when allpα pfor|α| k, then the space is denoted byWk,p,pα|α|≤k−1Ω.qθ|θ| ≤kis termed
the critical embedding exponent fromWk,pα|α|≤kΩtoLpΩ, ifq
θis the largest number of the
exponentpin whereDθu∈LpΩ, for allu∈Wpα
|α|≤kΩ, and the embedding is continuous.
For example, whenΩis bounded, the spaceX {u∈LkΩ|k ≥1, Diu∈L2Ω,1 ≤
i ≤ n} with norm u ∇uL2 uLk is an anisotropic Sobolev space, and the critical
embedding exponents fromXtoLPΩareq
i21≤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:
1aαβx, η aαβx;
2aαβx, η 0, asα /β.
A2There is a constantM >0 such that
0≤M
|α||β|m
aαβx,0ξαξβ≤
|α||β|m
aαβ
x, ηξαξβ
≤M−1
|α||β|m
aαβx,0ξαξβ.
A3There are functionsGix, η i0,1, . . . , nwithGix,0 0, for all 1≤i≤n, such
that
|γ|m−1
gγ
x,uDγu n
i1
DiGi
x,uG0
x,u. 3.7
A4There is a constantC >0 such that
C|ξ|2≤
|α||β|m−1
-dαβxξαξβ−
1 2
n
i1
Dibiαβxξαξβ
.
,
C
|λ|≤m−1
signpληλpλ−f1≤
|θ|≤m−2
gθ
x, ηηθG0
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
ηβSβ
1
⎤ ⎦,
gγ
x, η≤C
⎡ ⎣
|β|≤m−2
ηβSβ1
⎤ ⎦,
3.9
where 1 ≤ Sβ < qβ/2, 1 ≤ Sβ < qβ, qβ is a critical embedding exponent from
Wm2,pλ−1,|λ|≤m−1ΩtoLPΩ. LetXbe defined by2.27andX
1be the completion ofX under the norm
v1
⎡ ⎣$
Ω
⎛ ⎝
|α||β|m
aαβx,0DαvDβv
|γ|m−1
|Dγv|2
⎞ ⎠dx
$
∂Ω
Nm−1
i1
|hix|
⎛ ⎝Nm−1
j1
CBijDγjv
⎞ ⎠
2
ds
⎤ ⎥ ⎦
1/2
|γ|≤m−2
signpγDγvLpγ,
andX2be the completion ofXwith the norm
vvWm,p vWm,2 ⎡ ⎣$
∂Ω
|γ|m−1
|Dγv|2ds
⎤ ⎦
1/2
, 3.11
wherep≥max{2, qβ/qβ−Sβ, 2qβ/qβ−2Sβ}.
u∈X1is a weak solution of3.3, if for anyv∈X2, we have
$
Ω
⎡ ⎣
|α||β|m
aαβ
x,uDβuDαv
|α|m,|γ|m−1
bαγxDγuDαv
|γ||θ|m−1
dγθ
x,uDθuDγv
|λ|≤m−1
gλ
x,uDλv−fv
⎤ ⎦dx
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CBijDγju
⎞ ⎠
⎛ ⎝Nm−1
j1
CBijDγjv
⎞ ⎠ds0.
3.12
Theorem 3.1. Under the conditionsA1–A5, iff∈Lp0
Ω,1/p01/p0 1, then the problem
3.3has a weak solution inX1.
Proof. Denote by Au, v the left part of 3.12. It is easy to verify that the inner product
Au, vdefines a bounded mappingA:X1 → X2∗by the conditionA5. Letu∈X, byA2–A4, one can deduce that
Au, u ≥
$
Ω
⎡ ⎣M
|α||β|m
aαβx,0DαuDβuC
|γ|m−1
|Dγu|2C
|θ|≤m−2
Dθupθ
⎤ ⎦dx
1
2
Nm−1
i1
⎡ ⎢ ⎣ $
B i
− $
C i
hix
⎛ ⎝Nm−1
j1
CBijxDγju
⎞ ⎠
2⎤
⎥ ⎦ds−
$
Ω
/
fuf10dx.
3.13
Noticing thathi|B i >0,hi|
C i ≤0,
B
i ∪
C
i ∂Ω, by H ¨older and Young inequalitiessee13,
from3.13we can get
Au, u ≥0, ∀u∈X, uX1 large enough. 3.14
Ones can easily show that the mappingA:X1 → X2∗is weakly continuous. Here we omit
y
x ΣB
2
ΣB
1∩ΣB2
θ
ΣB
1
Figure2
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:
∂3u
∂x3
∂3u
∂y3 −Δuu
3fx, y, x, y∈Ω⊂R2, 3.15
whereΩis an unit ball inR2, seeFigure 2 The odd term matrix is
Bx, y
1
nx 0
0 ny
2
1
x 0
0 y
2
. 3.16
It is easy to see that
B
1
{x∈∂Ω|nxx >0} −π
2 < θ <
π
2
!
,
B
2
+x∈∂Ω|nyy >0
,
{0< θ < π}.
3.17
The boundary value condition associated with3.15is
u|∂Ω 0,
∂u ∂x
B
1 ∂u
∂xcosθ,sinθ 0, − π
2 < θ <
π
2,
∂u ∂x
B
2 ∂u
∂xcosθ,sinθ 0, 0< θ < π.
ApplyingTheorem 3.1, if f ∈ L4/3Ω, then the problem3.15–3.18has a weak solution
u∈W1,2Ω.
4.
W
m,p-Solutions of Degenerate Elliptic Equations
We start with an abstract regularity result which is useful for the existence problem of
Wm,pΩ-solutions of degenerate quasilinear elliptic equations of order 2m. LetX, X
1, X2be the spaces defined in Definition 2.6, andY be a reflective Banach space, at the same time
Y →X1.
Lemma 4.1. Under the hypotheses ofLemma 2.7, there exists a sequence of{un} ⊂X, un u0in
X1such thatGun, un0. Furthermore, if, we can derive thatuY < C,Cis a constant, then the
solutionu0ofGu0belongs toY.
In the following, we give some existence theorems ofWm,p-solutions for the boundary
value conditions4.3–4.5of higher-order degenerate elliptic equations. First, we consider the quasilinear equations
3
Au
|α||β|m,|γ|m−1
−1mDαa αβ
x,Du3 Dβub
αγxDγu
|γ|≤m−1
−1|γ|Dγgγ
x,Du3 fx, x∈Ω,
4.1
whereDu3 {Dαu}
|α|≤m−1. Now, we consider the following problem
3
Aufx, x∈Ω, 4.2
3
Du|∂Ω 0, 4.3
Nm−1
j1
CijBxDλju|B i 0,
λjm−1, 1≤i≤Nm−1, 4.4
Nm
j1
CM ij x,0D
αj−δ kju·n
kj|M
i 0, ∀δkj≤α j,
αjm, 1≤i≤N
m, δkj
⎧ ⎪ ⎨ ⎪ ⎩0 , . . . ,1
kj
, . . . ,0
⎫ ⎪ ⎬ ⎪ ⎭.
4.5
The boundary value condition associated with4.1is given by4.3–4.5. Suppose
B1The condition3.6holds, and there is a continuous function λx ≥ 0 onΩsuch that
λx|ξ|2m≤
|α||β|m
aαβx,0ξαξβ, ∀ξ∈Rn, 4.6
whereξαξα1
1 · · ·ξ
αn
n ,α α1, . . . , αn.
B2 Ω {x ∈ Ω | λx 0}is a measure zero set inRn, and there is a sequence of subdomainsΩkwith cone property such thatΩk⊂⊂Ω/Ω, Ωk⊂Ωk1and∪kΩk
Ω/Ω.
B3The positive definite condition is
C
|λ|≤m−1
|ξλ|pλ−f1≤
|θ|≤m−1
gθx, ξξθ−
1 2
n
i1
|γ||α|m−1
Dibiξαξγ, 4.7
whereCis a constant,p0>1, pλ>1 orpλ0 for 1≤ |λ| ≤m−1, f1∈L1Ω.
B4The structure conditions are
aαβx, ξ≤C,
gγx, ξ≤C⎡⎣
|θ|≤m−1
|ξθ|Sθ1
⎤ ⎦,
4.8
where Cis a constant, 0 ≤ Sθ < qθ,qθ is the critical embedding exponent from
W|pλλ|≤m−1ΩtoLPΩ.
LetXbe defined by2.27andX31be the completion ofXwith the norm
u
⎡ ⎣$
Ω
|α||β|m
aαβx,0DαuDβu dx
⎤ ⎦
1/2
|α|≤m−1
signpαDαuLpα
⎡ ⎣Nm−1
i1
$
∂Ω
|hix|
⎛ ⎝Nm−1
j1
CB ijxDγ
j
u
⎞ ⎠ds
⎤ ⎦
1/2
.
Definition 4.2. u∈X31is a weak solution of4.2–4.5, if for anyv∈X2, the following equality holds:
$
Ω
⎡ ⎣
|α||β|m
aαβ
x,Du3 DβuDαv
|α|m,|γ|m−1
bαγxDγuDαv
|γ|≤m−1
gγ
x,Du3 Dγv−fv
⎤ ⎦dx
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CijBDγju
⎞ ⎠
⎛ ⎝Nm−1
j1
CijBDγjv
⎞ ⎠ds0.
4.10
Theorem 4.3. Under the assumptionsB1–B4, iff ∈Lp0
, then the problem and4.2–4.5has a weak solutionu∈X31. Moreover, if there is a real numberδ≥1, such that
$
Ω|λx|
−δdx <∞, 4.11
then the weak solutionu∈Wm,pΩ∩X31, p2δ/1δ.
Proof. According toLemma 4.1, it suffices to prove that there is a constantC >0 such that for anyu∈XXis as that inSection 3withAu, u3 0, we have
uWm,p ≤C, p
2δ
1δ. 4.12
From4.10we know
4 3
Au, u5
$
Ω
⎡ ⎣
|α||β|m
aαβ
x,Du3 DβuDαu
|α|m,|γ|m−1
bαγxDγuDαu
|γ|≤m−1
gγ
x,Du3 Dγu−fu
⎤ ⎦dx
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CijBDγju
⎞ ⎠
1/2
ds, x∈X1.
Due toB1andB3we have
4 3
Au, u5
$
Ω
⎡ ⎣
|α||β|m
aαβ
x,Du3 DβuDαu
n
i1
|α||γ|m−1
biαγxDγuDαδiu
|γ|≤m−1
gγ
x,Du3 Dγu−fu
⎤ ⎦dx
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CBijxDγju
⎞ ⎠ 2 ds $ Ω ⎡ ⎣
|α||β|m
aαβ
x,Du3 DβuDαu−1
2
n
i1
|α||γ|m−1
DibiαγxDγuDαu
|γ|≤m−1
gγ
x,Du3 Dγu−fu
⎤ ⎦dx
−Nm−1
i1
$
C i
hix
⎛ ⎝Nm−1
j1
CBijxDγju
⎞ ⎠ 2 ds ≥ $ Ω ⎡
⎣λx|∇u|2mC
|θ|≤m−1
Dθupθ
⎤ ⎦dx−
$
Ω
/
fuf10dx
1
2
Nm−1
i1
⎡ ⎢ ⎣ $ B i− C i
hix
⎛ ⎝Nm−1
j1
CijBxDγju
⎞ ⎠
2⎤
⎥
⎦ds. 4.14
Noticing thathi|B
i >0, hi|
C i ≤0,
B
i ∩
C
i ∂Ω, andf∈Lp0
consequently we have
ε
$
Ω|u|
p0dx $
Ω
6
C1fp0
f17dx
≥ $
Ω
/
fuf10dx≥
$
Ω
⎡
⎣λx|∇u|2mC
|θ|≤m−1
DθuPθ
⎤ ⎦dx,
4.15
where thepθ> 1 orpθ 0,pθis the critical embedding exponent fromW|pθθ|≤m−1ΩtoLpΩ.
By the reversed H ¨older inequalitysee14
$
Ωλx|∇u|
2m≥
8$
Ω|λx|
−δdx
9−1/δ8$
Ω|∇u|
Then we obtain
C≥
$
Ω
⎡
⎣λx|∇u|2mC
|θ|≤m−1
DθuPθ
⎤
⎦dx. 4.17
From4.15and4.17, the estimates4.12follows. This completes the proof.
Next, we consider a quasilinear equation
|α||β|m,|γ|m−1
−1mDαaαβx,uDβubαβxDγu
|γ|≤m−1
−1|γ|Dγgγx,u fx, x∈Ω,
4.18
whereu{u, . . . , Dmu}.
Suppose that the following holds.
B4There is a real numberδ≥1 such that
$
Ω|λx|
−δdx <∞. 4.19
B5The structural conditions are
aαβ
x, η≤C,
gγx, ξ≤C⎡⎣
|θ|≤m−1
|ξθ|Sγθ
|α|m
|ξα|tγ 1
⎤
⎦,
4.20
where C is a constant, 0≤Sγθ <qγ−1/qγqθ, 0 ≤tγ < pqγ−1/qγ, p2δ/1δ,qγ, qθ
are the critical embedding exponents fromW|pλλ≤m−1|ΩtoLqΩ.
Theorem 4.4. Let the conditions B1–B3 and B4,B5be satisfied. If f ∈ Lp0
Ω, then the problem4.2–4.5has a weak solutionu∈Wm,pΩ∩X3
1, p2δ/1δ.
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.
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