| 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 {\it 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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