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Geometric Analysis and Partial Differential Equations Seminar
May 28, 2014 @ 10:30 am - 11:30 pm
Dr Daniel Hauer (University of Sydney)
Uniform convergence of solutions to elliptic equations on domains with shrinking holes
In this talk I want to present new results which I established during my postdoc year at the University of Sydney under the supervision of EN Dancer and D Daners. I consider solutions of the Poisson equation on a family of domains with holes shrinking to a point. This kind of domain convergence is very singular. Assuming Robin or Neumann boundary conditions on the boundary of the holes, I 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.