Advances in Differential Equations

Uniform convergence of solutions to elliptic equations on domains with shrinking holes

E.N. Dancer, Daniel Daners, and Daniel Hauer

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We consider solutions of the Poisson equation on a family of domains with holes shrinking to a point. Assuming Robin or Neumann boundary conditions on the boundary of the holes, we show that the solution converges uniformly to the solution of the Poisson equation on the domain without the holes. This is in contrast to Dirichlet boundary conditions where there is no uniform convergence. The results substantially improve earlier results on $L^p$-convergence. They can be applied to semi-linear problems.

Article information

Adv. Differential Equations, Volume 20, Number 5/6 (2015), 463-494.

First available in Project Euclid: 30 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations 35B25: Singular perturbations 35B45: A priori estimates


Daners, Daniel; Hauer, Daniel; Dancer, E.N. Uniform convergence of solutions to elliptic equations on domains with shrinking holes. Adv. Differential Equations 20 (2015), no. 5/6, 463--494.

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