Dirichlet problem for divergence form elliptic equations with discontinuous coefficients
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Dirichlet problem for divergence form elliptic equations with discontinuous coefficients Sara Monsurrò and Maria Transirico* *
Correspondence: [email protected] Dipartimento di Matematica, Università di Salerno, via Ponte Don Melillo, Fisciano (SA), 84084, Italy
Abstract We study the Dirichlet problem for linear elliptic second order partial differential equations with discontinuous coefficients in divergence form in unbounded domains. We establish an existence and uniqueness result and we prove an a priori bound in Lp , p > 2. MSC: 35J25; 35B45; 35R05 Keywords: elliptic equations; discontinuous coefficients; a priori bounds
1 Introduction We are interested in the Dirichlet problem ⎧ ◦ , ⎨u ∈W (Ω), ⎩Lu = f ,
(.)
f ∈ W –, (Ω),
where Ω is an unbounded open subset of Rn , n ≥ , and L is a linear uniformly elliptic second order differential operator with discontinuous coefficients in divergence form L=–
n n ∂ ∂ ∂ aij + dj + bi + c. ∂x ∂x ∂x j i i i,j= i=
(.)
If Ω is bounded, this problem is classical in literature and has been deeply analyzed taking into account various kinds of hypotheses on the coefficients (for more details see, for instance, [–]). Considering unbounded domains, as far as we know, the first work on this subject goes back to [], where Bottaro and Marina provide, for n ≥ , an existence and uniqueness result for the solution of problem (.) assuming that aij ∈ L∞ (Ω), bi , di ∈ Ln (Ω), c–
n
i, j = , . . . , n, i = , . . . , n,
(di )xi ≥ μ,
μ ∈ R+ .
(.) c ∈ Ln/ (Ω) + L∞ (Ω),
(.) (.)
i=
In this order of ideas, various generalizations have been performed still maintaining hypotheses (.) and (.) but weakening the condition (.). Indeed in [], where the case © 2012 Monsurrò and Transirico; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Monsurrò and Transirico Boundary Value Problems 2012, 2012:67 http://www.boundaryvalueproblems.com/content/2012/1/67
Page 2 of 12
n ≥ is considered, bi , di and c are supposed to satisfy assumptions as those in (.), but just locally. Successively in [], for n ≥ , further improvements have been carried on since bi , di and c are in suitable Morrey-type spaces with lower summabilities. In [–] we also find the bound uW , (Ω) ≤ Cf W –, (Ω) ,
(.)
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 Ω, and if f ∈ L (Ω) ∩ L∞ (Ω), then there exists a constant C, whose dependence is completely described, such that uLp (Ω) ≤ Cf Lp (Ω) ,
(.)
for any bounded solution u of (.) and for every p ∈ ], +∞[. Here, in the same framework but replacing
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