Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $Ω$ in ${\mathbb{R}}^N$. We consider deformations $ϕ(Ω)$ of $Ω$ obtained by means of a locally Lipschitz homeomorphism $ϕ$ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $ϕ$. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $ϕ$. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains.

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