| Abstract: |
| We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $\\mathbb{R}^{n}$ indexed by a small parameter $\\varepsilon$. The domains depend on $\\varepsilon$ only within a ball of radius proportional to $\\varepsilon$ and, as $\\varepsilon$ tends to zero, they converge in a self-similar way to a domain with a conical boundary singularity. We construct an expansion of the solution as a series of real positive powers of $\\varepsilon$, and prove that it is not just an asymptotic expansion as $\\varepsilon\\to0$, but that, for small values of $\\varepsilon$, it converges normally in the Sobolev space $H^{1}$. A planar version of such problem has been previously investigated by the authors with the so called Functional Analytic Approach, based on integral representations obtained through layer potentials.
Here, instead, we choose a different technique that allows us to relax all regularity assumptions. We forgo boundary layer potentials and instead exploit expansions in terms of eigenfunctions of the Laplace-Beltrami operator on the intersection of the cone with the unit sphere. The basis for our analysis is a two-scale cross-cutoff ansatz for the solution that has similarities with the Maz'ya-Nazarov-Plamenevskij construction of a multiscale system for the asymptotic expansion of solutions of boundary value problems on domains singularly perturbed near singular points of the boundary. |
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