SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 58-87, January 2020.
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 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.

中文翻译：

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随机介质中的亥姆霍兹方程：适定性和先验界

SIAM / ASA不确定性量化杂志，第8卷，第1期，第58-87页，2020年1月。
我们证明了适定性结果和Helmholtz方程$ \ nabla \ cdot（A \ nabla u）+ k ^ 2 nu = -f $的解的先验界，并置于$ \ mathbb {R} ^ d $中或在星形Lipschitz障碍物的外部，对于一类随机$ A $和$ n，$随机数据$ f $，以及所有$ k> 0 $。$ A $和$ n $的特定类以及障碍物的条件确保了该问题几乎可以肯定地解决。这些是最初的适定性结果，是任意大的k $以及$ A $和$ n $独立于$ k $的随机Helmholtz方程的先验界限。这些结果是通过将确定性$ A $和$ n $的Helmholtz方程的最新范围与一般参数（即，本文提出的非特定于Helmholtz方程的变量，以证明线性椭圆型随机PDE的变分公式的先验界和适定性。我们强调这些一般结果不依赖于Lax-Milgram定理或Fredholm理论，因为它们都不适用于Helmholtz方程的随机变分形式。