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1995 Integral representations and $L^\infty$ bounds for solutions of the Helmholtz equation on arbitrary open sets in $\mathbb{R}^2$ and $\mathbb{R}^3$
Wenzheng Xie
Differential Integral Equations 8(3): 689-698 (1995).

Abstract

We establish sharp $L^{\infty}$ bounds for functions defined on arbitrary open sets in $\Bbb R^2$ and $\Bbb R^3$, which vanish on the boundary and have $L^2$ Laplacians. All functions corresponding to the best possible constants are explicitly given. The proof is based on integral representations using the Green's function for the Helmholtz equation in arbitrary domains.

Citation

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Wenzheng Xie. "Integral representations and $L^\infty$ bounds for solutions of the Helmholtz equation on arbitrary open sets in $\mathbb{R}^2$ and $\mathbb{R}^3$." Differential Integral Equations 8 (3) 689 - 698, 1995.

Information

Published: 1995
First available in Project Euclid: 23 May 2013

zbMATH: 0824.35013
MathSciNet: MR1306585

Subjects:
Primary: 35C15
Secondary: 35J05

Rights: Copyright © 1995 Khayyam Publishing, Inc.

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Vol.8 • No. 3 • 1995
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