On the solvability of the first nonlocal boundary value problem for an elliptic equation
type: Item
instance of: scholarly article
title: On the solvability of the first nonlocal boundary value problem for an elliptic equation
zbMATH Open document ID: 1182.35097
DOI: 10.1134/S001226610906007X
author: A. K. Ratyni
published in: Differential Equations
publication date: 2009-10-08T00:00:00Z
review text: The author studies problems of existence, uniqueness and sign-definiteness of the classical solutions of the problem \[ Lu(x)= f(x), \quad x \in D; \qquad u(x)- \beta(x) u(\sigma x)= \psi(x), \quad x \in S, \] where \(D\) is a bounded domain in \(\mathbb R^n\) with boundary \(S\), \(\sigma \) is a single-valued continuous mapping, \(\beta, \psi \in C(S)\). A lot of papers deals with the solvability of boundary value problems for elliptic equations with nonlocal boundary conditions in various function spaces. In this note more general objects are considered, we point out the problem of the existence of solutions in situations in which the support of a nonlocal data meet the boundary of the domain. Under natural assumptions on the smoothness of \(\beta, \sigma\) and the coefficients of \(L\), the treated problem is Fredholm provided that either \(\sigma\) has no attractors on \(S\), or \(\sigma \) generates an attractor on \(S\) and the spectral radius of the ``generalized weighted shift'' operator occurring in the boundary condition \[ (Az)(x)= \beta (x) z(\sigma x), \quad x \in S, \] is less than unity. The author obtains semieffective conditions, in terms of test functions, for the unique solvability of the problem.
reviewed by: Maria Alessandra Ragusa
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 5611090
zbMATH Keywords: elliptic equations
zbMATH Keywords: nonlocal boundary value problem
zbMATH Keywords: Hölder spaces
Wikidata QID: Q115252743
MaRDI profile type: Publication
cites work: Q3746080
cites work: Q3031245
cites work: On the stability of index of nonlocal elliptic problems
cites work: On Feller semigroups generated by elliptic operators with integro-differential boundary conditions
cites work: Q3218622
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description: scientific article; zbMATH DE number 5611090