On analysis of problems of mathematical physics with non-Lipschitz boundaries

Abstract

We review recent advances in solving problems of mathematical physics on domains with irregular boundaries in Rn. We distinguish two frameworks: a measure-free approach in the image of the trace operator spaces for extension domains and an L2-approach depending on a d-upper regular boundary measure. In both cases, the domains can have boundaries with different Hausdorff dimensions inside the interval (n -- 2, n). The generalization of the Poincar{\'e}-Steklov/Dirichlet-to-Neumann operator for these two contexts is given. To illustrate the established convergence of spectral problems for elliptic operators with Robin boundary conditions, we give a numerical example of the stability of localized eigenfunctions, using results of M. Graffin.