Uniform estimates for a family of Poisson problems: `rounding off the corners'

2024/06/07 by Daniel, Benoît, Labrunie, Simon, Nistor, Victor
#35J15 #35R01 #46E35 #58J32 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.04773

Abstract

We prove uniform solvability estimates for certain families of elliptic problems posed in a bounded family of domains (for example, a sequence that converges to another domain). We provide uniform estimates both in weighted and in usual Sobolev spaces. When the limit domain is a polygon and the other domains are smooth, our results amount to rounding off'' the corners of the limit domain. The technique of proof is based on a suitable conformal modification of the metric, which makes the union of the domains a manifold with boundary and relative bounded geometry.

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