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Dirichlet and Neumann problems for planar domains with parameter

Authors: Florian Bertrand and Xianghong Gong
Journal: Trans. Amer. Math. Soc. 366 (2014), 159-217
MSC (2010): Primary 31A10, 45B05, 30C35, 35B30, 32H40
Published electronically: May 21, 2013
MathSciNet review: 3118395
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Abstract: Let $ \Gamma (\cdot ,\lambda )$ be smooth, i.e. $ \mathcal C^\infty $, embeddings from $ \overline {\Omega }$ onto $ \overline {\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 $ \overline \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 $ \overline \Omega \times I$. Our main result is proved for suitable Hölder 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 $ \overline {\mathbb{D}}$, the closure of the unit disc, onto $ \overline {\Omega ^\lambda }$ such that $ \Gamma $ is smooth on $ \overline {\mathbb{D}}\times I$ and real analytic at $ (\sqrt {-1},0)\in \overline {\mathbb{D}}\times I$, but for every family of Riemann mappings $ R(\cdot ,\lambda )$ from $ \overline {\Omega ^\lambda }$ onto $ \overline {\mathbb{D}}$, the function $ R(\Gamma (z,\lambda ),\lambda )$ is not real analytic at $ (\sqrt {-1},0)\in \overline {\mathbb{D}}\times I$.

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Additional Information

Florian Bertrand
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria

Xianghong Gong
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Keywords: Dirichlet and Neumann problems, Kellogg's theorem with parameter, domains with parameter, integral equations with parameter
Received by editor(s): October 31, 2011
Published electronically: May 21, 2013
Additional Notes: The research of the second author was supported in part by NSF grant DMS-0705426.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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