Estimates for Solutions to Anisotropic Elliptic Equations with Zero Order Term
Estimates for solutions to homogeneous Dirichlet problems for a class of elliptic equations with zero order term in the form \(L(u)= g(x, u) + f(x)\) , where the operator L fulfills an anisotropic elliptic condition, are established. Such estimates are ob
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Abstract Estimates for solutions to homogeneous Dirichlet problems for a class of elliptic equations with zero order term in the form L(u) = g(x, u) + f (x), where the operator L fulfills an anisotropic elliptic condition, are established. Such estimates are obtained in terms of solutions to suitable problems with radially symmetric data, when no sign conditions on g are required. Keywords Anisotropic symmetrization · A priori estimate · Anisotropic Dirichlet problems
1 Introduction We are concerned with a comparison result via symmetrization for solutions to a class of anisotropic Dirichlet problems, whose prototype can be written as follows ⎧ N ⎪ ⎪ ¯ ⎪ αi (|∂xi u| pi −2 ∂xi u)xi = c(x)|u| p−2 u + f (x) ⎨ ⎪ ⎪ ⎪ ⎩
in Ω (1)
i=1
u=0
on ∂Ω,
A. Alberico Istituto per le Applicazioni del Calcolo “M. Picone”, Sez. Napoli, C.N.R., Via P. Castellino 111, 80131 Napoli, Italy e-mail: [email protected] G. di Blasio Dipartimento di Matematica e Fisica, Seconda Università degli Studi di Napoli, Via Vivaldi 43, 81100 Caserta, Italy e-mail: [email protected] F. Feo (B) Dipartimento di Ingegneria, Centro Direzionale Isola C4, Università degli Studi di Napoli “Pathenope”, 80143 Napoli, Italy e-mail: [email protected] © Springer International Publishing Switzerland 2016 F. Gazzola et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer Proceedings in Mathematics & Statistics 176, DOI 10.1007/978-3-319-41538-3_1
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where Ω is a bounded, smooth open subset of R N , N ≥ 2, αi > 0 for i = 1, . . . , N , 1 ≤ p1 , . . . , p N < ∞ such that their harmonic mean p¯ is greater than 1, and f belongs to a suitable Lebesque space. In the last years anisotropic problems have been largely studied by many authors (see e.g. [6, 10, 15, 20–22, 27]). We observe that when pi = p = 2 for every i = 1, . . . , N , the principal part operator in problem (1) coincides with the so-called pseudo-Laplacian operator, whereas when pi = 2 for every i = 1, . . . , N the operator coincides with the usual Laplacian. Symmetrization methods in comparison results for solutions to isotropic elliptic problems with a zero lower order term were used in several papers (see e.g. [5, 18, 19, 28], and bibliography therein). In the quoted papers the authors require either sign assumptions on c(x) or not. In the same spirit of [19], we are interested in studying the anisotropic problem (1) when no sign assumption is made on c(x). Using symmetrization techniques we want estimate a solution to problem (1) with the solution to an appropriate symmetrized problem which takes into account also of the influence of the zero order term. More precisely, we compare the decreasing rearrangement u ∗ of a solution u to problem (1) with the decreasing rearrangement v∗ of the solution v to the following symmetrized problem ⎧ ¯ ¯ ∇v = c|v| p−2 v+ f ⎨ −div Λ|∇v| p−2 ⎩
in Ω on ∂Ω .
v=0
(2)
In (2) the datum f is the spherically symmetric decreasing rearrangement of
f , Ω is the ball centered at th
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