# The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances

@article{Graham2018TheHE,
title={The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances},
author={I. Graham and O. R. Pembery and E. Spence},
journal={arXiv: Analysis of PDEs},
year={2018}
}
• Published 2018
• Mathematics
• arXiv: Analysis of PDEs
• We consider a variety of boundary value problems (BVPs) for the heterogeneous Helmholtz equation, i.e. the equation $\nabla\cdot(A \nabla u ) + k^2 n u =-f$ where both $A$ and $n$ are functions of position. We prove new a priori bounds on the solution under conditions on $A$, $n$, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in $k$, $A$, $n$, and geometric parameters of the domain. We then show that these a priori bounds hold when $A$ and $n$ are… CONTINUE READING
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