\(W^{2,2}\)-solvability of the Dirichlet problem for a class of elliptic equations with discontinuous coefficients
type: Item
instance of: scholarly article
title: \(W^{2,2}\)-solvability of the Dirichlet problem for a class of elliptic equations with discontinuous coefficients
zbMATH Open document ID: 1395.35084
DOI: 10.4171/RLM/820
author: Flavia Giannetti
author: Gioconda Moscariello
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2018-07-13T00:00:00Z
review text: Summary: We study the Dirichlet problem for the second order elliptic equation \[ -\sum\limits_{i,j=1}^Na_{ij}(x)\frac{\partial^2u}{\partial x_i\partial x_j}=f(x) \] in a bounded regular domain \(\Omega\subset\mathbb{R}^N,\;N>2\). We assume that \(f\in L^{2}\) and that the coefficients \(a_{ij}\) are measurable and bounded functions with the first derivatives in the Marcinkiewicz class weak-\(L^{N}\) and having a sufficiently small distance to \(L^{\infty}\). Under these assumptions we prove the solvability of the problem in \(W^{2,2}\cap W_0^{1,2^\ast}\), where \(2^\ast=\frac{2N}{N-2}\). An higher integrability result for the gradient of the solution is achieved when \(f\in L^p,\;p>2\).
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 6903550
zbMATH Keywords: elliptic equation
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/rlm/820
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description: scientific article; zbMATH DE number 6903550