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The Helmholtz equation in random media
Abstract
We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation ∇ (A∇u)+k 2nu = -f, posed either in 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 emphasize that these general results do not rely on either the Lax-Milgram theorem or Fredholm theory, since neither is applicable to the stochastic variational formulation of the Helmholtz equation.