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PDE comparison principles for Robin problems

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  23 September 2021

Jeffrey J. Langford
Jeffrey J. Langford*
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA, USA
*
e-mail: jeffrey.langford@bucknell.edu
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Abstract

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data have been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger convex means. Sending one of the Robin parameters to $+\infty $, we obtain mixed Robin/Dirichlet comparison results in shells. We prove similar results on balls and prove a comparison principle on generalized cylinders with mixed Robin/Neumann boundary conditions.

Keywords

Symmetrizationcomparison theoremsPoisson’s equationRobin boundary conditions

MSC classification

Primary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 35B51: Comparison principles

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Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 108 - 139
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Copyright
© Canadian Mathematical Society 2021

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