Dirichlet problem for divergence form elliptic equations with discontinuous coefficients

Monsurrò and Transirico Boundary Value Problems 2012, 2012:67http://www.boundaryvalueproblems.com/content/2012/1/67

R E S E A R C H Open Access

Dirichlet problem for divergence form ellipticequations with discontinuous coefficientsSara Monsurrò and Maria Transirico*

*Correspondence:[email protected] di Matematica,Università di Salerno, via Ponte DonMelillo, Fisciano (SA), 84084, Italy

AbstractWe study the Dirichlet problem for linear elliptic second order partial di!erentialequations with discontinuous coe"cients in divergence form in unboundeddomains. We establish an existence and uniqueness result and we prove an a prioribound in Lp, p > 2.MSC: 35J25; 35B45; 35R05

Keywords: elliptic equations; discontinuous coe"cients; a priori bounds

1 IntroductionWe are interested in the Dirichlet problem

!"

#u !

"W,(!),

Lu = f , f ! W –,(!),(.)

where ! is an unbounded open subset of Rn, n # , and L is a linear uniformly ellipticsecond order differential operator with discontinuous coefficients in divergence form

L = –n$

i,j=

"

"xj

%aij

"

"xi+ dj

&+

n$

i=bi

"

"xi+ c. (.)

If ! is bounded, this problem is classical in literature and has been deeply analyzed tak-ing into account various kinds of hypotheses on the coefficients (for more details see, forinstance, [–]).

Considering unbounded domains, as far as we know, the first work on this subject goesback to [], where Bottaro and Marina provide, for n # , an existence and uniquenessresult for the solution of problem (.) assuming that

aij ! L$(!), i, j = , . . . , n, (.)

bi, di ! Ln(!), i = , . . . , n, c ! Ln/(!) + L$(!), (.)

c –n$

i=(di)xi # µ, µ ! R+. (.)

In this order of ideas, various generalizations have been performed still maintaining hy-potheses (.) and (.) but weakening the condition (.). Indeed in [], where the case

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n # is considered, bi, di and c are supposed to satisfy assumptions as those in (.), butjust locally. Successively in [], for n # , further improvements have been carried on sincebi, di and c are in suitable Morrey-type spaces with lower summabilities.

In [–] we also find the bound

%u%W ,(!) & C%f %W –,(!), (.)

where the dependence of the constant C on the data of the problem is fully determined.More recently, in [], supposing that the coefficients of lower-order terms are as in

[] for n # and as in [] for n = , we showed that, for a sufficiently regular set ! , andif f ! L(!) ' L$(!), then there exists a constant C, whose dependence is completelydescribed, such that

%u%Lp(!) & C%f %Lp(!), (.)

for any bounded solution u of (.) and for every p ! ], +$[.Here, in the same framework but replacing the classical hypothesis of sign (.) by the

less common one

c –n$

i=(bi)xi # µ, µ ! R+, (.)

we establish two kinds of results for the solution of (.). First of all, we provide an exis-tence and uniqueness theorem, then, taking into account an additional assumption on theregularity of the boundary of ! , we prove the analogue of (.).

Let us briefly survey the way these results are achieved. In Section , we introduce thetools needed in the sequel. The definitions and some features of the Morrey-type spacesare given and some functions us, related somehow to the solution of the problem andto the coefficients of the operator, are described, together with some specific properties.Section is devoted to the solvability of problem (.). We start proving, by means of theabove mentioned functions us, the estimate in (.) that leads also to the uniqueness atonce. Then, in view of well-known results of the operator theory, we get the existenceverifying that L is a Fredholm operator with zero index. In the last section, we prove theclaimed Lp-estimate. This is done by means of a technical lemma, exploiting again thefunctions us, which allows us to conclude.

Considering the case p = , we notice that, as a consequence of (.), the bound (.) istrue under both sign hypotheses even supposing no regularity on the boundary of ! .

We believe that the two estimates (.), obtained under the different sign assumptions,combined together should permit to prove, by means of a duality argument, that (.)holds true actually for any p ! ], +$[, considering one of the hypotheses (.) or (.) ata time.

For further studies of the Dirichlet problem for linear elliptic second order differentialequations with discontinuous coefficients in divergence form in unbounded domains werefer the reader also to [–].

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2 ToolsThis section is devoted to the definitions and to some fundamental properties of theMorrey-type spaces where the coefficients of lower-order terms of our operator belong,and of some functions us related to the solution of the problem and to all the coefficientsof the operator (see the proofs of Theorem . and Lemma . for more details on thisaspect) that are indispensable tools in the sequel.

Given an unbounded open subset ! of Rn, n # , we denote by #(!) the $ -algebra ofall Lebesgue measurable subsets of ! . For any E ! #(!), %E is its characteristic functionand E(x, r) is the intersection E ' B(x, r) (x ! Rn, r ! R+), where B(x, r) is the open ballcentered in x and with radius r.

For q ! [, +$[ and & ! [, n[, the space of Morrey type Mq,&(!) is the set of all thefunctions g in Lq

loc(!̄) such that

%g%Mq,&(!) = sup'! ],]

x!!

'–&/q%g%Lq(!(x,' )) < +$,

endowed with the norm above defined. Moreover, Mq,&" (!) denotes the closure of C$

" (!)in Mq,&(!). These functional spaces generalize the classical notion of Morrey spaces tothe case of unbounded domains and were introduced in [] (we refer also to [] wherefurther characteristics are considered).

For the reader’s convenience, in the next lemma we recall some results of [] and [, ]concerning the multiplication operator

u !"

W,(!) ( gu ! L(!), (.)

where the function g belongs to suitable spaces of Morrey type.

Lemma . If g ! Mq,&(!), with q > and & = if n = , and q ! ], n] and & = n – q ifn > , then the operator in (.) is bounded and there exists a constant c ! R+ such that

%gu%L(!) & c%g%Mq,&(!)%u%W ,(!) )u !"

W,(!), (.)

with c = c(n, q).Moreover, if g ! Mq,&

" (!), then the operator in (.) is also compact.

Now, let us deal with the above mentioned functions us. They were employed for thefirst time in [] and were studied in the framework of Morrey-type spaces in [].

For h > k # , we define the functions of the real variable t

Gk$(t) =

!''"

''#

t – k if t > k, if – k & t & k,t + k if t < –k,

(.)

and

Gkh(t) = Gk$(t) – Gh$(t). (.)

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Lemma . Let g ! Mq,&o (!), u !

"W ,(!) and ( ! R+. Then there exist r ! N and

k, . . . , kr ! R, with = kr < kr– < · · · < k < k = +$, such that set

us = Gksks– (u), s = , . . . , r, (.)

one has u, . . . , ur !"

W,(!) and

u = u + · · · + ur , (.)

us & uus, s = , . . . , r, (.)

|us| & |u|, s = , . . . , r, (.)

uxi (us)xj = (us)xi (us)xj , s = , . . . , r, i, j = , . . . , n, (.)

u(us)xi = (us + · · · + ur)(us)xi , s = , . . . , r, i = , . . . , n, (.)

%g%supp(us)x%Mq,&(!) & (, s = , . . . , r, (.)

r & c, (.)

with c = c((, q,%g%Mq,&(!)) positive constant.

Proof The proofs of the properties (.), (.), (.), (.) and (.) can be found in [].Inequality (.) is an immediate consequence of (.).Considering (.), observe that in the case s = it is a trivial consequence of (.).Thus let us fix s ! N and such that & s & r. As already proved in [] and in [], in the

case of unbounded domains, one has

(Gksks– (u)

)xi

= G*ksks– (u)uxi , a.e. in ! , i = , . . . , n.

This, together with (.) and (.), gives

supp(us)xi +*

x ! ! s.t. ks < |u| < ks–, uxi ,= +

, (.)

i = , . . . , n.On the other hand, by definition,

supp uh +*

x ! ! s.t. |u| # kh+

, h = , . . . , r. (.)

Combining (.) and (.), we conclude that

supp uh ' supp(us)xi = -,

h = , . . . , s – , i = , . . . , n. Hence by (.) we get (.). !

3 Existence and uniqueness resultLet ! be an unbounded open subset of Rn, n # .

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We are interested in the study of the following Dirichlet problem in ! :!"

#u !

"W,(!),

Lu = f , f ! W –,(!),(.)

where L is a second order linear differential operator in divergence form

L = –n$

i,j=

"

"xj

%aij

"

"xi+ dj

&+

n$

i=bi

"

"xi+ c, (.)

satisfying the following hypotheses on the leading coefficients:!''"

''#

aij ! L$(!), i, j = , . . . , n,

.) > :n$

i,j=aij*i*j # )|* | a.e. in ! ,)* ! Rn. (h)

Considering the coefficients of lower-order terms, we suppose that!'''''"

'''''#

bi, di ! Mt,&(!), di – bi ! Mt,&o (!), i = , . . . , n,

c ! Mt,&(!),with t > and & = if n = ,with t ! ], n/] and & = n – t if n > ,

(h)

!''"

''#

c –n$

i=(bi)xi # µ, µ = constant > ,

in the sense of distributions on ! .(h)

We associate to L the bilinear form

a(u, v) =,

!

- n$

i,j=(aijuxi + dju)vxj +

- n$

i=biuxi + cu

.

v.

dx, (.)

u, v !"

W,(!).As a consequence of Lemma ., a is continuous on

"W,(!)/

"W,(!); and therefore,

the operator L :"

W,(!) ( W –,(!) is continuous too.

Theorem . Under hypotheses (h)-(h), problem (.) is uniquely solvable and its solu-tion u satisfies the estimate

%u%W ,(!) & C%f %W –,(!), (.)

where C is a constant depending on n, t, ) , µ, %di – bi%Mt,&(!), i = , . . . , n.

Proof We start proving estimate (.) that yields also to the uniqueness of the solutionat once. Successively, in view of classical results concerning operator theory, to get theexistence, it will be enough to verify that L is a Fredholm operator with zero index.

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Let us, for s = , . . . , r, be the functions of Lemma . corresponding to a solution u of(.), to g = /n

i= |di – bi| and to a positive real number ( that will be specified in the sequel.By a well-known characterization of the space W –,(!), we have

f = f –n$

i=(fi)xi , fk ! L(!), k = , . . . , n.

Thus, if we take us as a test function in the variational formulation of problem (.), bysimple calculations and (.) and (.), we obtain

,

!

fus dx +n$

i=

,

!

fi(us)xi dx

= a(u, us)

=,

!

0 n$

i,j=aijuxi (us)xj +

n$

i=

(diu(us)xi + biuxi us

)+ cuus

1

dx

=,

!

0 n$

i,j=aijuxi (us)xj +

n$

i=bi(uus)xi + cuus +

n$

i=(di – bi)u(us)xi

1

dx

=,

!

0 n$

i,j=aij(us)xi (us)xj +

n$

i=bi(uus)xi + cuus

+n$

i=(di – bi)

- r$

h=suh

.

(us)xi

1

dx.

Hypotheses (h) and (h) together with (.) give then

,

!

fus dx +n$

i=

,

!

fi(us)xi dx # )

,

!

(us)x dx + µ

,

!

(us) dx

–,

!

r$

h=s|uh|

n$

i=|di – bi|

22(us)xi22dx.

(.)

On the other hand, by the Hölder inequality, the embedding results contained in Lem-ma . and using hypothesis (h) and (.), one has that there exists a constant c ! R+such that

,

!

r$

h=s|uh|

n$

i=|di – bi|

22(us)xi22dx

&r$

h=s

33|uh|g%supp(us)x33

L(!)33(us)x

33L(!)

& c

r$

h=s%uh%W ,(!)%g%supp(us)x%Mt,&(!)%us%W ,(!)

& (c%us%W ,(!)

r$

h=s%uh%W ,(!),

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with c = c(n, t).Hence, set

µ = min{),µ},

by (.) we get

µ%us%W ,(!) & %f%L(!)%us%L(!) +

n$

i=%fi%L(!)

33(us)xi33

L(!)

+ (c%us%W ,(!)

r$

h=s%uh%W ,(!).

Thus, choosing ( = µc

we have

%us%W ,(!) & µ

%f %W –,(!) +

r$

h=s%uh%W ,(!),

for s = , . . . , r.If we rewrite the last inequality for s = r and we estimate %ur%W ,(!), then for s = r –

and we estimate %ur–%W ,(!) and so on, we get by substituting that

%us%W ,(!) & r–s+

µ%f %W –,(!),

for s = , . . . , r.Therefore, taking into account (.), we conclude that

%u%W ,(!) &r$

s=%us%W ,(!) &

(r –

) µ

%f %W –,(!).

This, together with (.), ends the proof of the bound in (.).Now, as it was already mentioned, it only remains to show that the operator

L : u !"

W,(!) ( Lu ! W –,(!)

is a Fredholm operator with zero index.To this aim, set + = /n

i=(di – bi) and denote by + u, u !"

W ,(!), the element ofW –,(!) given by

+ u : v !"

W,(!) (,

!

+ uv dx,

which is well defined in view of Lemma ..Then, consider the problem

!"

#u !

"W,(!),

Lu + )

+ u = f , f ! W –,(!).(.)

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Clearly, if we show that (.) has a unique solution, we end our proof, since in this casethe operator L can be seen as a sum between a Fredholm operator with zero index and acompact operator; and therefore, it is a Fredholm operator with zero index itself.

Indeed, we explicitly observe that the operator

u !"

W,(!) ( + u ! W –,(!)

is compact, since, by hypothesis (h) and Lemma ., it is obtained as a composition be-tween the compact operator

u !"

W,(!) ( + /u ! L(!)

and the bounded one

v ! L(!) ( + /v ! W –,(!),

where + /v, v ! L(!), is the element of W –,(!) defined by

+ /v : w !"

W,(!) (,

!

+ /vw dx.

To get the existence and uniqueness of the solution of problem (.), we want to makeuse of Lax-Milgram Lemma. Thus let us consider the bilinear form associated to it

a(u, v) + )

,

!

+ uv dx, u, v !"

W,(!). (.)

The continuity of the form (.) can be easily obtained by Lemma .. Considering thecoercivity, for every u !

"W,(!), in view of hypotheses (h) and (h), one has

a(u, u) =,

!

n$

i,j=aijuxi uxj dx +

,

!

n$

i=

(bi

(u)

xi+ cu)dx

+,

!

n$

i=(di – bi)uuxi dx

# )%ux%L(!) + µ%u%

L(!) +,

!

n$

i=(di – bi)uuxi dx.

On the other hand, Hölder and Young inequalities give that

,

!

n$

i=|di – bi||u||uxi |dx & )

%ux%L(!) +

)

n$

i=

33(di – bi)u33

L(!)

and therefore,

a(u, u) + )

,

!

+ u dx # min4

)

,µ5%u%

W ,(!).

This concludes the proof of Theorem .. !

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4 An a priori bound in Lp

Here we want to prove, for a sufficiently regular datum f , a Lp-a priori estimate, p > , fora bounded solution of problem (.).

To this aim, we require a further assumption on the boundary of ! :

! has the uniform C-regularity property. (h)

Moreover, a technical lemma below is needed. We note that the proof of Lemma .follows the idea of the one of the estimate (.). However, in this case, there are somespecific arguments that need to be explicitly treated.

Let us be the functions of Lemma . corresponding to a fixed u !"

W,(!) ' L$(!), tog = /n

i= |di – bi| and to a positive real number ( to be specified in the proof of Lemma ..The following result holds true:

Lemma . Let a be the bilinear form in (.). Under hypotheses (h)-(h), there exists aconstant C ! R+ such that

,

!

|us|p–((us)x + u

s)

dx & Cr$

h=sa(u, |uh|p–uh

), s = , . . . , r,)p ! ], +$[, (.)

where C depends on s, r, ) , µ.

Proof Let u, g , ( and us, for s = , . . . , r, be as above specified. Since u ! L$(!), by defini-tion of us and by Lemma ., the functions us !

"W ,(!) ' L$(!). Therefore, in view of

hypothesis (h), Lemma . in [] applies giving that |us|p–us !"

W,(!) for any p > .Thus, we can take |us|p–us as a test function in (.), obtaining by (.) that

a(u, |us|p–us

)=

,

!

0 n$

i,j=aijuxi

(|us|p–us

)xj

+n$

i=bi

(|us|p–usu

)xi

+ c|us|p–usu +n$

i=(di – bi)u

(|us|p–us

)xi

1

dx

=,

!

0

(p – )|us|p–n$

i,j=aij(us)xi (us)xj

+n$

i=bi

(|us|p–usu

)xi

+ c|us|p–usu

+ (p – )|us|p–un$

i=(di – bi)(us)xi

1

dx.

If we set

µ = min{),µ}

and

Hs(u) = |us|p–((us)x + (us)), (.)

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by hypotheses (h) and (h) and in view of (.), one has

µ

,

!

Hs(u) dx & a(u, |us|p–us

)

+ (p – ),

!

g|us|p–|u|(us)x dx.(.)

On the other hand, by (.), (.) and (.), using the Hölder inequality, we get that thereexists a constant c ! R+, such that

,

!

g|u||us|p–(us)x dx &,

!

g|u||u|p/–|us|p/–(us)x dx

& c

r$

h=s

,

!

g|uh|p/|us|p/–(us)x dx

& c33|us|p/–(us)x

33L(!)

r$

h=s

33g|uh|p/%supp(us)x33

L(!),

with c = c(r, p).Thus, using hypothesis (h), by Lemma . and (.), we obtain

,

!

g|u||us|p–(us)x dx

& c33|us|p/–(us)x

33L(!)%g%supp(us)x%Mt,&(!)

r$

h=s

33|uh|p/33W ,(!)

& c(33|us|p/–(us)x

33L(!)

r$

h=s

33|uh|p/33W ,(!),

(.)

with c = c(r, p, n, t).Now, we observe that explicit calculations give

33|uh|p/33W ,(!) & p

%,

!

Hh(u) dx&/

, h = s, . . . , r. (.)

Hence, putting together (.), (.) and (.), we get,

!

Hs(u) dx & µ

a(u, |us|p–us

)

+ cµ

(

%,

!

Hs(u) dx&/ r$

h=s

%,

!

Hh(u) dx&/

,

with c = c(r, p, n, t).Thus, by Young inequality,

,

!

Hs(u) dx & µ

a(u, |us|p–us

)+ c

µ(

%,

!

Hs(u) dx&/- r$

h=s

,

!

Hh(u) dx./

& µ

a(u, |us|p–us

)+ c

µ

-,

,

!

Hs(u) dx + (

,

r$

h=s

,

!

Hh(u) dx.

,

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with c = c(r, p, n, t).Choosing , = µ

cand ( = µ

c0

we have

,

!

Hs(u) dx & µ

a(u, |us|p–us

)+

r$

h=s

,

!

Hh(u) dx, (.)

s = , . . . , r.If we rewrite the last inequality for s = r, then for s = r – and take into account the

estimate of6!

Hr(u) dx obtained in the previous step, and so on, we conclude our proof.Indeed, we get

,

!

Hs(u) dx & Cr$

h=sa(u, |uh|p–uh

),

with C = C(s, r,µ). !

We are finally in position to prove the above mentioned Lp-bound.

Theorem . Assume that the hypotheses (h)-(h) are satisfied. If f is in L(!) ' L$(!)and the solution u of (.) is in

"W,(!) ' L$(!), then

%u%Lp(!) & C%f %Lp(!) )p ! ], +$[,

where C is a constant depending on n, t, p, ) , µ, %di – bi%Mt,&(!), i = , . . . , n.

Proof Fix p ! ], +$[. If we consider the functions us, s = , . . . , r, corresponding to thesolution u, to g and ( as in Lemma ., easy computations together with (.) give that

,

!

|u|p dx & c

r$

s=

,

!

|us|p dx

with c = c(r, p).Thus, by (.), one has

,

!

|u|p dx & c

r$

s=Cs

r$

h=sa(u, |uh|p–uh

)& c

r$

s=a(u, |us|p–us

),

with Cs = Cs(s, r,),µ) and c = c(r, p,),µ).Hence by (.) and Hölder inequality, we get

%u%pLp(!) & c

r$

s=

,

!

f |us|p–us dx

& rc

,

!

|f ||u|p– dx & rc%f %Lp(!)%u%p–Lp(!).

This concludes the proof, in view of (.). !

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Competing interestsThe authors declare that they have no competing interests.

Author’s contributionsThe authors conceived and wrote this article in collaboration and with the same responsibility. Both of them read andapproved the final manuscript.

AcknowledgementThe authors would like to thank anonymous referees for a careful reading of this article and for valuable suggestions andcomments.

Received: 27 February 2012 Accepted: 15 June 2012 Published: 28 June 2012

References1. Chicco, M: An a priori inequality concerning elliptic second order partial di!erential equations of variational type.

Matematiche 26, 173-182 (1971)2. Gilbarg, D, Trudinger, NS: Elliptic Partial Di!erential Equations of Second Order, 2nd edn. Springer, Berlin (1983)3. Ladyzhenskaja, OA, Ural’tzeva, NN: Equations aux Derivèes Partielles de Type Elliptique. Dunod, Paris (1966)4. Miranda, C: Alcune osservazioni sulla maggiorazione in L) delle soluzioni deboli delle equazioni ellittiche del

secondo ordine. Ann. Mat. Pura Appl. 61, 151-169 (1963)5. Stampacchia, G: Le problème de Dirichlet pour les équations elliptiques du second ordre à coe"cients discontinus.

Ann. Inst. Fourier (Grenoble) 15, 151-169 (1966)6. Trudinger, NS: Linear elliptic operators with measurable coe"cients. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 27, 265-308

(1973)7. Bottaro, G, Marina, ME: Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati. Boll.

Unione Mat. Ital. 8, 46-56 (1973)8. Transirico, M, Troisi, M: Equazioni ellittiche del secondo ordine a coe"cienti discontinui e di tipo variazionale in aperti

non limitati. Boll. Unione Mat. Ital, B 2, 385-398 (1988)9. Transirico, M, Troisi, M, Vitolo, A: Spaces of Morrey type and elliptic equations in divergence form on unbounded

domains. Boll. Unione Mat. Ital, B 9, 153-174 (1995)10. Monsurrò, S, Transirico, M: A Lp-estimate for weak solutions of elliptic equations. Abstr. Appl. Anal. (2012).

doi:10.1155/2012/37617911. Chicco, M, Venturino, M: Dirichlet problem for a divergence form elliptic equation with unbounded coe"cients in an

unbounded domain. Ann. Mat. Pura Appl. 178, 325-338 (2000)12. Lions, PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les

domaines non bornés. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 78, 205-212 (1985)13. Lions, PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les

domaines non bornés II. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 79, 178-183 (1985)14. Caso, L, D’Ambrosio, R, Monsurrò, S: Some remarks on spaces of Morrey type. Abstr. Appl. Anal. (2010).

doi:10.1155/2010/24207915. Cavaliere, P, Longobardi, M, Vitolo, A: Imbedding estimates and elliptic equations with discontinuous coe"cients in

unbounded domains. Matematiche 51, 87-104 (1996)16. Stampacchia, G: Equations elliptiques du second ordre à coe"cients discontinus. Séminaire de mathématiques

supérieures, Université de Montréal, 4e session, été 1965. Les presses de l’Université de Montréal, Montreal (1966)17. Caso, L, Cavaliere, P, Transirico, M: Solvability of the Dirichlet problem in W2,p for elliptic equations with discontinuous

coe"cients in unbounded domains. Matematiche 57, 287-302 (2002)

doi:10.1186/1687-2770-2012-67Cite this article as: Monsurrò and Transirico: Dirichlet problem for divergence form elliptic equations withdiscontinuous coefficients. Boundary Value Problems 2012 2012:67.