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

Abstract

We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\nabla \cdot (A\nabla u)+k^{2}nu=-f$, 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.

36 pages, 2 figures