The Helmholtz equation in random media: well-posedness and a priori bounds

TL;DR

This paper establishes the first well-posedness and a priori bounds for the stochastic Helmholtz equation with random media and data, valid for all wave numbers, by combining deterministic bounds and novel general arguments.

Contribution

It introduces new methods to prove well-posedness and bounds for the stochastic Helmholtz equation without relying on classical theorems like Lax-Milgram or Fredholm theory.

Findings

First well-posedness results for stochastic Helmholtz with large wave numbers

A priori bounds for solutions in random media

Applicable to problems in unbounded and obstacle domains

Abstract

We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\nabla\cdot(A\nabla u) + k^2 n u = -f$, posed either in $\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular class of $A$ and $n$ and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$ and for $A$ and $n$ varying independently of $k$. These results are obtained by combining recent bounds on the Helmholtz equation for deterministic $A$ and $n$ and general arguments (i.e. not specific to the Helmholtz equation) presented in this paper for proving a priori bounds and well-posedness of variational formulations of linear elliptic stochastic PDEs. We emphasise that these general results do not rely on either the Lax-Milgram theorem or Fredholm theory, since neither are applicable to the stochastic variational formulation of the Helmholtz equation.