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Uniform convergence of solutions to elliptic equations on domains with shrinking holes. (English) Zbl 1319.35026

In this article, solutions of the Poisson equation on a variety of domains including holes shrinking to a point are considered. The authors prescribe on the boundary either Robin or Neumann conditions and prove that the solution converges uniformly to the solution of the Poisson equation on the same domain but without the holes. This is very surprising, since this uniform convergence is not valid for the case with Dirichlet boundary conditions. The given results improve earlier results dealing with \(L^p\)-convergence. Hence, they might be applicable to semi-linear problems.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
35B45 A priori estimates in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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Full Text: Euclid