A condition for the complete continuity of operators arising in nonlocal problems for elliptic equations.
type: Item
instance of: scholarly article
title: A condition for the complete continuity of operators arising in nonlocal problems for elliptic equations.
zbMATH Open document ID: 1040.35020
author: A. K. Gushchin
published in: Doklady Mathematics
publication date: 2004-06-15T00:00:00Z
review text: The author studies the Dirichlet problem for the uniformly elliptic operator \[ D_i(a_{ij}(x)D_ju)=0, \quad x\in Q,\qquad u|_{\partial Q}=Bu+u_0 \] where \((Bu)(x)=\int_{\overline Q}b(x,y)u(y)\,d\nu_x(y),\) \(x\in \partial Q.\) The domain \(Q\) supposed to be bounded with a smooth boundary and the normal to \(\partial Q\) satisfies the Dini condition, the coefficients are measurable and bounded in \(Q\) and satisfy the Dini condition on the boundary. It is introduced a functional space and is described the basic properties of the solutions of the problem under consideration with a boundary function from \(L^2(\partial Q)\) in terms of inclusion in this space. It is established a condition for the complete continuity of integral operators from the space introduced to \(L^2(\partial Q)\) that are determined by measures close to the Carleson measures. This makes it possible to substantially strengthen results on Fredholmness of a wide class of nonlocal problems.
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J67
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 2074619
zbMATH Keywords: elliptic equations
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: complete continuity
reviewed by: Lubomira Softova
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2074619