Solving a Dirichlet problem on unbounded domains via a conformal transformation. (English) Zbl 07867547
Summary: In this paper, we solve the \(p\)-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter \(p\). We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.
MSC:
| 31E05 | Potential theory on fractals and metric spaces |
| 30L99 | Analysis on metric spaces |
| 49Q05 | Minimal surfaces and optimization |
| 26A45 | Functions of bounded variation, generalizations |
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