Dirichlet and Neumann problems for planar domains with parameter

TL;DR

This paper studies the smooth dependence of harmonic solutions to Dirichlet and Neumann problems on smoothly varying planar domains with a parameter, revealing non-analyticity in Riemann mappings despite smooth embeddings.

Contribution

It establishes the smoothness of harmonic solutions with respect to domain deformations and demonstrates the non-analyticity of Riemann mappings under certain smooth embeddings.

Findings

Harmonic solutions depend smoothly on domain parameters.

Existence of smooth embeddings with non-analytic Riemann mappings.

Regularity results in suitable H"older spaces for parameter-dependent problems.

Abstract

Let $\Gamma(\cdot,\lambda)$ be smooth, i.e.\, $\mathcal C^\infty$, embeddings from $\bar{\Omega}$ onto $\bar{\Omega^{\lambda}}$, where $\Omega$ and $\Omega^\lambda$ are bounded domains with smooth boundary in the complex plane and $\lambda$ varies in $I=[0,1]$. Suppose that $\Gamma$ is smooth on $\bar\Omega\times I$ and $f$ is a smooth function on $\partial\Omega\times I$. Let $u(\cdot,\lambda)$ be the harmonic functions on $\Omega^\lambda$ with boundary values $f(\cdot,\lambda)$. We show that $u(\Gamma(z,\lambda),\lambda)$ is smooth on $\bar\Omega\times I$. Our main result is proved for suitable H\"older spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings $\Gamma(\cdot,\lambda)$ from $\bar{\mathbb D}$, the closure of the unit disc, onto $\bar{\Omega^\lambda}$ such that $\Gamma$ is smooth on $\bar{\mathbb D}\times I$ and real analytic at $(\sqrt{-1},0)\in\bar{\mathbb D}\times I$, but for every family of Riemann mappings $R(\cdot,\lambda)$ from $\bar{\Omega^\lambda}$ onto $\bar{\mathbb D}$, the function $R(\Gamma(z,\lambda),\lambda)$ is not real analytic at $(\sqrt{-1},0)\in\bar{\mathbb D}\times I$.