Regularity of weak solutions to certain degenerate elliptic equations

Regularity of weak solutions to certain degenerate elliptic equations

Albo Carlos Cavalheiro

Abstract. In this article we establish the existence of higher order weak derivatives of weak solutions of Dirichlet problem for a class of degenerate elliptic equations.

Keywords: degenerate elliptic equations, weighted Sobolev spaces Classification: Primary 35J70; Secondary 35J25

1. Introduction

In this paper we study the existence of higher order weak derivatives (see Theorem 3.8) of weak solutions of degenerate elliptic equationsLu=g−divf~, whereLis an elliptic operator

(1.1) Lu=−

Xn i,j=1

Dj(aij(x)Diu)(x)− Xn i=1

bi(x)Diu(x)

whose coefficients a_ij and b_i are measurable, real-valued functions, and whose coefficient matrix A= (a_ij) is symmetric and satisfies the degenerate ellipticity condition

(1.2) λω(x)|ξ|²≤

Xn i,j=1

a_ij(x)ξ_iξ_j ≤Λω(x)|ξ|²

for allξ∈Rⁿand almost allx∈Ω⊂Rⁿon a bounded open set Ω,ω is a weight function,λand Λ are positive constants.

2. Definitions and basic results

By a weight we shall mean a locally integrable function ω on Rⁿ such that 0 < ω(x)<∞ for a.e. x ∈Rⁿ. Every weight ω gives rise to a measure on the measurable subsets ofRⁿthrough integration. This measure will also be denoted byω. Thusω(E) =R

Eω dxfor measurable sets E⊂Rⁿ.

The author was partially supported by CNPq Grant 476040/2004-03.

Definition 2.1. Let Ω⊂Rⁿ be open and letω be a weight. For 1< p <∞, we defineL^p(Ω, ω), the Banach space of all measurable functions f defined on Ω for which

kfk_L^p_(Ω,ω)= Z

Ω|f(x)|^pω(x)dx 1/p

<∞.

Definition 2.2. Let 1≤p <∞. A weightω belongs to the Muckenhoupt class Ap if there is a constantC=Cp,ω such that

1

|B|

Z

Bω dx 1

|B|

Z

Bω^−1/(p−1)dx p−1

≤C (if 1< p <∞) 1

|B|

Z

B

ω dx

ess sup

B

1 ω

≤C (if p= 1),

for every ballB⊂Rⁿ, where|B|is then-dimensional Lebesgue measure ofB. The infimum over all constantsC is called “Ap-constant ofω”.

Example 2.3. The functionω(x) =|x|^α, x∈Rⁿ, is a weightAp if and only if

−n < α < n(p−1) (see [6, Chapter 15]).

Remark 2.4. Ifω ∈Ap, 1≤p <∞, then sinceω^−1/(p−1) is locally integrable whenp >1, and 1/ωis locally bounded, whenp= 1, we haveL^p(Ω, ω)⊂L¹_loc(Ω) for every open set Ω. If Ω is bounded, one obtains in the same way thatL^p(Ω, ω)⊂ L¹(Ω). It thus makes sense to talk about weak derivatives of functions inL^p(Ω, ω).

Definition 2.5. Let Ω⊂Rⁿ be a bounded open set, 1 ≤ p < ∞ and k be a nonnegative integer. Suppose that the weightω ∈ Ap. We define the weighted Sobolev spaceW^k,p(Ω, ω) as the set of functionsu∈L^p(Ω, ω) with weak deriva- tivesD^αu∈L^p(Ω, ω) for|α| ≤k. The norm ofuinW^k,p(Ω, ω) is given by

kuk_Wk,p(Ω,ω)= X

0≤|α|≤k

Z

Ω|D^αu|^pω dx 1/p

.

We also defineW₀^k,p(Ω, ω) as the closure ofC₀^∞(Ω) inW^k,p(Ω, ω).

If Ω ⊂ Rⁿ is open, k ≥ 1, 1 ≤ p < ∞ and ω ∈ A_p then C^∞(Ω) is dense in W^k,p(Ω, ω) (see Corollary 2.1.6 in [8]). The spaces W^k,p(Ω, ω) are Banach spaces.

In this paper we use frequently the following two theorems.

Theorem 2.6(Muckenhoupt Theorem). Letω be a weight inRⁿ and [M(f)](x) = sup

Q∋x

1

|Q|

Z

Q

|f(y)|dy

be the Hardy-Littlewood maximal function. Then forp >1, M :L^p(Rⁿ, ω)−→

L^p(Rⁿ, ω)is continuous (that is,kM fk_Lp(Rⁿ,ω)≤C_Mkfk_Lp(Rⁿ,ω)) if and only if ω∈A_p. The constantC_M is called Muckenhoupt constant andC_M depends only onn,pand theA_p-constant of ω.

Proof: See [7] or [4, Corollary 4.3].

Theorem 2.7 (Weighted Sobolev inequality). Let Ωbe a bounded open set in Rⁿ, 1< p <∞andω ∈Ap. Then there exist constantsC_Ω andδ positive such that for allϕ∈C₀^∞(Ω)and ksatisfying1≤k≤ _n−1ⁿ +δ,

kϕk_Lkp(Ω,ω)≤C_Ωk|▽ϕ|k_L^p_(Ω,ω)

where C_Ω may be taken to depend only on n, the Ap constant of ω, pand the diameter ofΩ.

Proof: See Theorem 1.3 of [2].

Definition 2.8. We say thatu∈W^1,2(Ω, ω) is a weak solution of the equation Lu=g−

Xn i=1

D_if_i, with g ω,f_i

ω ∈L²(Ω, ω) if

B(u, ϕ) = Xn i=1

Z

Ω

f_iD_iϕ+ Z

Ω

gϕ dx, ∀ϕ∈W₀^1,2(Ω, ω) where

B(u, ϕ) = Z

Ω

Xn i,j=1

a_ijD_iuD_jϕ− Xn i=1

b_iϕD_iu

dx.

Theorem 2.9. LetLbe the operator(1.1)satisfying(1.2)and|b_i(x)| ≤C₁ω(x) inΩ. Assume thatψ∈W^1,2(Ω, ω),g/ω∈L²(Ω, ω),f_i/ω∈L²(Ω, ω)andω∈A2. Then the Dirichlet problem

Lu=g− Xn i=1

Difi

u−ψ∈W₀^1,2(Ω, ω) has a unique solutionu∈W^1,2(Ω, ω)and

kuk_W^1,2_(Ω,ω)≤C

kg/ωk_L²_(Ω,ω)+kf_j/ωk_L2(Ω,ω)+kψk_W1,2(Ω,ω)

. Proof: It is consequence of the Lax-Milgram Theorem and the proof follows the

lines of Theorem 2.2. of [2].

3. Differentiability of weak solutions

In this section we prove that weak solutions u ∈ W^1,2(Ω, ω) of the equa- tion Lu = g are twice weakly differentiable and D_iju ∈ L²(Ω^′, ω) (that is, u∈W^2,2(Ω^′, ω),∀Ω^′⊂⊂Ω).

Definition 3.1. Letube a function on a bounded open set Ω⊂Rⁿand denote by e_ithe unit coordinate vector in thex_idirection. We define the difference quotient ofuat xin the direction e_i by

(3.1) ∆^h_ku(x) = u(x+he_k)−u(x)

h , (0<|h|<dist(x, ∂Ω)).

Lemma 3.2. Let Ω^′⊂⊂Ω and 0 < |h| < dist(Ω^′, ∂Ω). If u, v ∈ L²_loc(Ω, ω), supp(v)⊂Ω^′ andgis a measurable function with |g(x)| ≤Cω(x), then

(a) ∆^h_k(uv)(x) =u(x+he_k)∆^h_kv(x) +v(x)∆^h_ku(x), with1≤k≤n;

(b) R

Ωg(x)u(x)∆^−h_k v(x)dx=−R

Ωv(x)∆^h_k(gu)(x)dx;

(c) ∆^h_k(D_jv)(x) =D_j(∆^h_kv)(x).

Proof: The proof of this lemma follows trivially from Definition 3.1.

Definition 3.3. Letωbe a weight inRⁿ. We say thatωis uniformlyAp in each coordinate if

(a) ω∈A_p(Rⁿ);

(b) ω_i(t) =ω(x₁, . . . , x_i−1, t, x_i+1, . . . , x_n) is inA_p(R), forx₁, . . . , x_i−1, x_i+1, . . . , xn a.e., 1≤i≤n, withAp constant ofω_i bounded independently of x₁, . . . , x_i−1, x_i+1, . . . , xn.

Example 3.4. Letω(x, y) =ω₁(x)ω₂(y), withω₁(x) =|x|^1/2andω₂(y) =|y|^1/2. We have thatω is uniformlyA₂ in each coordinate.

Lemma 3.5. Letu∈W^1,p(Ω, ω), p >1, and letω be a weight uniformlyA_p in each coordinate. Then for anyΩ^′⊂⊂Ωand0<|h|<dist(Ω^′, ∂Ω), we have (3.2) k∆^h_kuk_L^p_(Ω′,ω)≤CkD_kuk_L^p_(Ω,ω)

whereC= 2C_M, andC_M is the Muckenhoupt constant.

Proof: Case 1. Let us suppose initially thatu∈C^∞(Ω). We have,

∆^h_ku(x) = u(x+he_k)−u(x)

h = 1

h Z h

0 D_k(x+ζe_k)dζ

= 1 h

Z h 0

D_ku(x₁, . . . , x_k−1, x_k+ζ, x_k+1, . . . , x_n)dζ.

For 1≤k≤n, we define the functions G_k(x) =

D_ku(x), if x∈Ω 0, if x /∈Ω.

We have forx∈Ω^′⊂⊂Ω andhsatisfying 0<|h|<dist(Ω^′, ∂Ω),

|∆^h_ku(x)| ≤ 1

|h|

Z _h

0

|D_ku(x₁, . . . , x_k−1, x_k+ζ, x_k+1, . . . , x_n)|dζ

= 1

|h|

Z xk+h xk

|D_ku(x₁, . . . , x_k−1, t, x_k+1, . . . , x_n)|dt

= 1

|h|

Z _xk+h

xk

|G_k(x₁, . . . , x_k−1, t, x_k+1, . . . , xn)|dt

≤ 1

|h|

Z xk+h

xk−h |G_k(x1, . . . , x_k−1, t, x_k+1, . . . , xn)|dt

≤sup

h6=0

1

|h|

Z xk+h xk−h

|G_k(x₁, . . . x_k−1, t, x_k+1, . . . , x_n)|dt

≤2M(G^x_k^{1,...,xk−1,xk+1,..,xn})(xk),

where G^x_k^{1,...,xk−1,xk+1,...,xn}(xk) = G_k(x₁, . . . ,xk, . . . , xn). Consequently, using the notationdxd_k=dx₁. . . dx_k−1dx_k+1. . . dxn(where the hat indicates the term that must be omitted in the product) and by Theorem 2.6, we obtain

Z

Ω^′

|∆^h_ku(x)|^pω(x)dx

≤2^p Z

Ω^′

[M(G^x_k^{1,...,xk−1xk+1,...,xn})]^p(xk)ω(x₁, . . . , x_k, . . . , xn)dx

≤2^p Z

Rⁿ

[M(G^x_k^{1,...,xk−1,xk+1,...,xn})]^p(xk)ω(x₁, . . . , x_k, . . . , xn)dx₁. . . dx_k. . . dxn

= 2^p Z

Rⁿ⁻¹

Z

R

[M(G^x_k^{1,...,xk−1xk+1,...,xn})]^p(xk)ω(x₁, . . . ,xk, . . . x_n)dx_k dxd_k

≤2^p Z

Rⁿ⁻¹

C_M^p

Z

R

|G^x_k^{1,...,xk−1,xk+1,...,xn}(xk)|^pω(x₁, . . . ,xk, . . . , xn)dx_k dxd_k

= 2^pC_M^p Z

Rⁿ

|G_k(x)|^pω(x)dx

= 2^pC_M^p Z

Ω|D_ku(x)|^pω(x)dx,

whereC_M is independent ofx₁, . . . , x_k−1, x_k+1, . . . , x_nbecauseωis uniformlyA_p in each coordinate. Therefore

k∆^h_kuk_L^p_(Ω′,ω)≤CkD_kuk_L^p_(Ω,ω), where C= 2C_M.

Case 2. If u ∈ W^1,p(Ω, ω) then there exists a sequence {u_m}, u_m ∈ C^∞(Ω), Cauchy sequence in the normk.k_W1,p(Ω,ω). By Definition 2.5, we have that

um−→u and D_kum−→D_ku in L^p(Ω, ω).

Consequently, sinceω ∈Ap, there exists a subsequence{umj}such thatumj−→u a.e. This implies, for 0<|h|<dist(Ω^′, ∂Ω), that

∆^h_kumj −→∆^h_ku a.e.

We have that {∆^h_kumj} is a Cauchy sequence inL^p(Ω^′, ω), for any Ω^′⊂⊂Ω. In fact, using the first case, we have

k∆^h_ku_mr−∆^h_ku_msk_Lp(Ω^′,ω)=k∆^h_k(u_mr−u_ms)k_Lp(Ω^′,ω)

≤CkD_k(u_mr−u_ms)k_L^p_(Ω,ω)

=CkD_kumr−D_kumsk_Lp(Ω,ω)

−→0, as m_r, m_s−→ ∞.

Therefore, there existsg ∈ L^p(Ω^′, ω) such that ∆^h_ku_mj−→g in L^p(Ω^′, ω). Con- sequently, there exists a subsequence ∆^h_kum_jr−→g a.e. We can conclude that

∆^h_ku=g a.e. Hence

∆^h_kumj −→∆^h_ku in L^p(Ω^′, ω).

This implies that

k∆^h_kuk_L^p_(Ω′,ω)= lim

mj→∞k∆^h_kumjk_L^p_(Ω′,ω)

≤C lim

mj→∞kD_kumjk_L^p_(Ω,ω)

=CkD_kuk_L^p_(Ω,ω),

that is,k∆^h_kuk_L^p_(Ω′,ω)≤CkD_kuk_L^p_(Ω,ω). Lemma 3.6. Letu∈L^p(Ω, ω), 1< p <∞, ω ∈Ap and suppose there exists a constantC such that

(3.3) k∆^h_kuk_Lp(Ω^′,ω)≤C, k= 1,2, . . . , n

for anyΩ^′⊂⊂Ωand0<|h|<dist(Ω^′, ∂Ω) (withCindependent of h). Then there existsv∈L^p(Ω, ω)such thatD_ku=v in the weak sense, that is,u∈W^1,p(Ω, ω) andkD_kuk_L^p_(Ω,ω)≤C.

Proof: The proof of this lemma follows the lines of Lemma 7.24 in [5].

Remark 3.7. We use the notation D^k(Ω, ω) =

g∈W^k(Ω) : D^αg

ω ∈L²(Ω, ω), 0≤ |α| ≤k

, for k= 0,1,2. . . , whereW^k(Ω) denotes the linear space ofktimes weakly derivative functions. For k= 0, we haveg∈ D⁰(Ω, ω) ifg/ω∈L²(Ω, ω).

We are able now to prove the main result of this paper.

Theorem 3.8. Letu∈ W^1,2(Ω, ω)be a weak solution of the equationLu =g inΩ, and assume that

(a) g∈ D⁰(Ω, ω);

(b) ω is a weight uniformlyA₂ in each coordinate;

(c) |∆^h_ka_ij(x)| ≤ C₁ω(x), x ∈ Ω^′⊂⊂Ω a.e., 0 < |h| < dist(Ω^′, ∂Ω), with a constantC1 independent of Ω^′ andh;

(d) |b(x)| ≤Cω(x)a.e. inΩ, where b= (b1, . . . , bn).

Then for any subdomainΩ^′⊂⊂Ωwe haveu∈W^2,2(Ω^′, ω)and (3.4) kuk_W2,2(Ω^′,ω)≤C

kuk_W1,2(Ω,ω)+kg/ωk_L²_(Ω,ω) forC=C(n, C_M, λ,Λ, C₁, d^′), andd^′= dist(Ω^′, ∂Ω).

Proof: Sinceu∈W^1,2(Ω, ω) is a weak solution of the equationLu=g, we have by

(3.5) Z

Ω

Xn i,j=1

a_ij(x)D_iu(x)D_jv(x)dx− Z

Ω

Xn i=1

b_i(x)D_iu(x)v(x)dx

= Z

Ω

g(x)v(x)dx

for allv∈W₀^1,2(Ω, ω) (in particular forv∈C₀^∞(Ω)). Hence Z

Ω

Xn i,j=1

a_ij(x)D_iu(x)D_jv(x)dx

= Z

Ω

[g(x)v(x) + Xn i=1

b_i(x)D_iu(x)v(x)]dx.

In (3.5) let us replacev by ∆^−h_k v(1≤k≤n), withv∈C₀^∞(Ω), suppv⊂⊂Ω and

let|2h|<dist(suppv, ∂Ω). We then obtain

− Z

Ω

[g(x) +b_i(x)D_iu(x)]∆^−h_k v(x)dx

=− Z

Ω

a_ij(x)D_iu(x)D_j(∆^−h_k v(x))dx

=− Z

Ωa_ij(x)D_iu(x)∆^−h_k D_jv(x), dx (by Lemma 3.2(b))

= Z

Ω∆^h_k(a_ijD_iu)(x)D_jv(x)dx (by Lemma 3.2(a))

= Z

Ω

a_ij(x+he_k)∆^h_kD_iu(x) +D_iu(x)∆^h_ka_ij(x)

D_jv(x)dx

= Z

Ω

[h∆^h_ka_ij(x) +a_ij(x)]∆^h_kD_iu(x) +D_iu(x)∆^h_ka_ij(x)

D_jv(x)dx.

Consequently, (3.6)Z

Ωa_ij(x)D_jv(x)∆^h_kD_iu(x)dx=− Z

Ω[g(x) +b_i(x)D_iu(x)]∆^−h_k v(x)dx +

Z

Ω

∆^h_ka_ij(x)D_iu(x)D_jv(x)dx+ Z

Ω

h∆^h_ka_ij(x)∆^h_kD_iu(x)D_jv(x)dx

≤ Z

Ω

|g(x) +b_i(x)D_iu(x)||∆^−h_k v(x)|dx+ Z

Ω

|∆^h_ka_ij(x)||D_iu(x)||D_jv(x)|dx +|h|

Z

Ω

|∆^h_ka_ij(x)||∆^h_kD_iu(x)||D_jv(x)|dx

= I + II +C₁|h|

Z

Ω

ω(x)|∆^h_kD_iu(x)||D_jv(x)|dx.

Let us estimate the integrals I and II. Consideringf =g+b_iD_iu, by (a) and (d), we have

I = Z

Ω

|f||∆^−h_k v|dx

= Z

Ω

|f| ω

ω^1/2|∆^−h_k v|ω^1/2dx

≤ Z

Ω

f ω

2

ω dx 1/2Z

supp(v)

|∆^−h_k v|²ω dx 1/2

≤C_Mkf /ωk_L²_(Ω,ω) Z

Ω|D_kv|²ω dx 1/2

(by Lemma 3.5)

=C_M

kg/ωk_L²_(Ω,ω)+C₁kuk_W1,2(Ω,ω)

kD_kvk_L²_(Ω,ω).

II = Z

Ω

|∆^h_ka_ij||D_iu||D_jv|dx≤ Z

Ω

C1ω|D_iu||D_jv|dx

=C1

Z

Ω

|D_iu|ω^1/2|D_jv|ω^1/2dx

≤C₁ Z

Ω

|D_iu|²ω dx

1/2 Z

Ω

|D_jv|²ωdx 1/2

≤C₁kuk_W1,2(Ω,ω)kD_jvk_L²_(Ω,ω).

Replacing the estimates of I and II in (3.6), we get the estimate

(3.7)

Z

Ω

a_ij(x)∆^h_kD_iu(x)D_jv(x)dx

≤C

kuk_W1,2(Ω,ω)+kg/ωk_L²_(Ω,ω)

kDvk_L²_(Ω,ω) +C₁|h|

Z

Ω

ω(x)|∆^h_kD_iu(x)||D_jv(x)|dx.

We denote bya=kuk_W1,2(Ω,ω)+kg/ωk_L²_(Ω,ω).

Let Ω^′⊂⊂Ω. To proceed further let us take a function ψ ∈ C₀^∞(Ω), satisfying 0≤ψ≤1,ψ ≡1 in Ω^′ and withkDψk∞ ≤ 2/d^′, whered^′ = dist(Ω^′, ∂Ω), and setv=ψ²∆^h_ku(with|2h|<dist(suppψ, ∂Ω)). We have

D_jv= (2ψD_jψ)∆^h_ku+ψ²D_j(∆^h_ku).

Then we obtain Z

Ω

a_ij(x)ψ²D_j(∆^h_ku)D_i(∆^h_ku) + 2a_ij(x)ψD_jψ∆^h_ku∆^h_kD_iu dx

≤Cak2ψD_jψ∆^h_ku+ψ²D_j(∆^h_ku)k_L²_(Ω,ω) +C1|h|

Z

Ω

ω(x)|∆^h_kD_iu(x)||2ψD_jψ∆^h_ku+ψ²D_j(∆^h_ku)|dx

≤Ca

kψDjψ∆^h_kuk_L²_(Ω,ω)+kψ²Dj(∆^h_ku)k_L²_(Ω,ω) +C1|h|

Z

Ωω(x)|∆^h_kDiu(x)||2ψDjψ∆^h_ku+ψ²Dj(∆^h_ku)|dx.

Then Z

Ω

a_ij(x)[ψD_i(∆^h_ku)][ψD_j(∆^h_ku)]dx

≤Ca

kψD_jψ∆^h_kuk_L²_(Ω,ω)+kψ²D_j(∆^h_ku)k_L²_(Ω,ω) + 2

Z

Ω

|a_ij(x)||ψD_i∆^h_ku||D_jψ∆^h_ku|dx +C₁|h|

Z

Ω

ω(x)|∆^h_ku(x)||2ψD_jψ∆^h_ku+ψ²D_j(∆^h_ku)|dx.

By (1.2) we have|a_ij(x)| ≤Cω(x), and we can estimate the integral on the right hand side by

Z

Ω

|a_ij(x)||ψD_i(∆^h_ku)||D_jψ∆^h_ku|dx

≤C Z

Ω

|ψD_i(∆^h_ku)||D_jψ∆^h_ku|ω dx

≤C Z

Ω

|ψD_i(∆^h_ku)|²ω dx 1/2Z

Ω

|D_jψ∆^h_ku|²ω dx 1/2

=CkψD_i(∆^h_ku)k_L²_(Ω,ω)kD_jψ∆^h_kuk_L²_(Ω,ω). Hence, we obtain

(3.8) Z

Ω

a_ij(x)[ψD_j(∆^h_ku)][ψD_i(∆^h_ku)]dx

≤CakψD_jψ∆^h_kuk_L²_(Ω,ω) +Cakψ²D_j(∆^h_ku)k_L²_(Ω,ω)

+ 2CkψD_i(∆^h_ku)k_L²_(Ω,ω)kD_jψ∆^h_kuk_L²_(Ω,ω) +C₁|h|

Z

Ω

ω(x)|∆^h_kD_iu(x)||2ψD_jψ∆^h_ku+ψ²D_j(∆^h_ku)|dx.

Finally, the integral on the right hand side in (3.8) can be estimated Z

Ω

ω(x)|∆^h_kD_iu(x)||2ψD_jψ∆^h_ku+ψ²D_j(∆^h_ku)|dx

≤ Z

Ω

2ω(x)|∆^h_kD_iu(x)||ψD_jψ∆^h_ku|+ Z

Ω

ω(x)|∆^h_kD_iu||ψ²D_j(∆^h_ku)|dx

= 2 Z

Ω

ω(x)|ψ∆^h_kD_iu||D_jψ∆^h_ku|dx+ Z

Ω

ω(x)|ψ∆^h_kD_iu||ψD_j(∆^h_ku)|dx

≤2kψ∆^h_kD_iuk_L²_(Ω,ω)kD_jψ∆^h_kuk_L²_(suppψ,ω) +kψ∆^h_kD_iuk_L²_(Ω,ω)kψ∆^h_kD_juk_L²_(Ω,ω). Applying this result in (3.8), we obtain

Z

Ω

a_ij(x)[ψD_j∆^h_ku][ψD_i∆^h_ku]dx

≤CakψD_jψ∆^h_kuk_L²_(suppψ,ω) +Cakψ²D_j∆^h_kuk_L²_(Ω,ω)

+ 2CkψD_i∆^h_kuk_L²_(Ω,ω)kD_jψ∆^h_kuk_L²_(suppψ,ω) + 2C₁|h|kψ∆^h_kD_iuk_L²_(Ω,ω)kD_jψ∆^h_kuk_L²_(suppψ,ω) +C₁|h|kψ∆^h_kD_iuk_L²_(Ω,ω)kψ∆^h_kD_juk_L²_(Ω,ω). Consequently, by condition (1.2), we then have

Z

Ωaij(x)[ψDj(∆^h_ku)][ψDi(∆^h_ku)]dx≥λ Z

Ω|ψD(∆^h_ku)|²ω dx.

Denotingb=kψD(∆^h_ku)k_L²_(Ω,ω), we have

λb²≤CakD_jψ∆^h_kuk_L²_(Ω,ω)+Cab+ 2CbkD_jψ∆^h_kuk_L²_(Ω,ω) + 2C₁|h|bkD_jψ∆^h_kuk_L²_(suppψ,ω)+C₁|h|b².

Using the Young’s inequality

AB= (ε⁻¹A)(εB)≤ 1

2[(ε⁻¹A)²+ (εB)²], ∀ε >0 to estimateabandbkD_jψ∆^h_kuk_L²_(Ω,ω), we obtain

λb² ≤CakD_jψ∆^h_kuk_L²_(Ω,ω)+C

2ε⁻²a²+C 2ε²b² + 2Cε²b²+Cε⁻²

2 kD_jψ∆^h_kuk²_L2(Ω,ω)

+ 2C1|h|bkDjψ∆^h_kuk_L²_(Ω,ω)+C1|h|b²

≤CakD_jψ∆^h_kuk_L²_(suppψ,ω)+Cε⁻²

2 a²+Cε² 2 b² + 2C²ε²b²+Cε⁻²

2 kD_jψ∆^h_kuk²_L2(suppψ,ω)

+C1ε²|h|²b²+C1ε⁻²

2 kD_jψ∆^h_kuk²_L2(suppψ,ω)+C1|h|b². Chooseε >0 andhsuch that

C

2ε²+ 2Cε²≤λ/4 and |h|< λ/8C₁.

Then C

2ε²+ 2Cε²+C₁|h|²+C₁|h|

≤ λ 2

and we can use Lemma 3.5 to get

λb²≤CakDjψ∆^h_kuk_L²_(suppψ,ω) +C

2ε⁻²a²+λ

2b²+Cε⁻²kD_jψ∆^h_kuk²_L2(suppψ,ω)

≤CakDjψk∞k∆^h_kuk_L²_(suppv,ω)+C 2ε⁻²a² +λ

2b²+Cε⁻²kD_jψk²_∞k∆^h_kuk²_L2(suppv,ω)

≤CakD_jψk∞kD_kuk_L²_(Ω,ω)+C

2ε⁻²a²+λ 2b² +Cε⁻²kD_jψk²_∞kD_kuk²_L2(Ω,ω).

SincekD_kuk_L²_(Ω,ω)≤a, we have λ

2b²≤CkD_jψk∞a²+C

2ε⁻²a²+Cε⁻²kD_jψk²_∞a²

≤

CkD_jψk∞+C

2ε⁻²+Cε⁻²kD_jψk²_∞

a²

=Ca². Consequently, we obtain

b≤ 2C

λ 1/2

a.

Usingψ≡1 in Ω^′, we conclude that

k∆^h_k(Du)k_L²_(Ω′,ω)≤Ca, ∀k, 1≤k≤n,∀Ω^′⊂⊂Ω

with 0 < |h| < dist(Ω^′, ∂Ω) and h < λ/8C₁. By Lemma 3.6 we obtain Du ∈ W^1,2(Ω^′, ω). Therefore we have that u∈W^2,2(Ω^′, ω) and

kuk_W^2,2_(Ω′,ω)≤Ca=C

kuk_W^1,2_(Ω,ω)+kg/ωk_L²_(Ω,ω) .

By a straightforward induction argument, we can then conclude the following extension of Theorem 3.8.

Theorem 3.9. Letu∈ W^1,2(Ω, ω)be a weak solution of the equationLu =g inΩ, and assume that

(a) ω is a weight uniformlyA₂ in each coordinate;

(b) g∈ D^k(Ω, ω),k∈N,k≥1;

(c) there exist D^αa_ij a.e. and |∆^h_pD^αa_ij(x)| ≤ C₁ω(x), x ∈ Ω^′⊂⊂Ω, 0 ≤

|α| ≤k,1≤p≤n,0<|h|<dist(Ω^′, ∂Ω), with constantC₁ independent of Ω^′ andh;

(d) there exist D^αb_i a.e., 0 ≤ |α| ≤ k−1, and |D^αb_i(x)| ≤ C2ω(x), x ∈ Ω^′⊂⊂Ω.

Then for any subdomainΩ^′⊂⊂Ω, we haveu∈W^k+2,2(Ω^′, ω)and kuk_Wk+2,2(Ω^′,ω)≤C

kuk_W^1,2_(Ω,ω)+ X

0≤|α|≤k

kD^αg/ωk_L²_(Ω,ω)

forC=C(n, λ,Λ, C_M, C1, C2, d^′), and d^′ = dist(Ω^′, ∂Ω).

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State University of Londrina (Universidade Estadual de Londrina), Department of Mathematics (Departamento de Matem´atica), 86051-990 Londrina - PR - Brasil E-mail: [email protected]

(Received December 21, 2005,revised June 20, 2006)