## Abstract

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

Original language | English |
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Pages (from-to) | 58-87 |

Number of pages | 30 |

Journal | SIAM/ASA Journal on Uncertainty Quantification |

Volume | 8 |

Issue number | 1 |

Early online date | 9 Jan 2020 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- A priori bounds
- Helmholtz equation
- High frequency
- Nontrapping
- Random media
- Well-posedness

## ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics
- Applied Mathematics