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2016•SIAM Journal on Mathematical AnalysisOpen access

Sharp High-Frequency Estimates for the Helmholtz Equation and Applications to Boundary Integral Equations

Dean Baskin, Euan A. Spence, Jared Wunsch

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Abstract

We consider three problems for the Helmholtz equation in interior and exterior domains in $\mathbb{R}^d$ ($d=2,3$): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.

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What this paper is about

We consider three problems for the Helmholtz equation in interior and exterior domains in $\mathbb{R}^d$ ($d=2,3$): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.

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Available abstract

We consider three problems for the Helmholtz equation in interior and exterior domains in $\mathbb{R}^d$ ($d=2,3$): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.

Key concepts: Mathematics, Helmholtz equation, Mathematical analysis, Dirichlet distribution, Integral equation, Neumann boundary condition, Electric-field integral equation, Dirichlet problem

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