Vol. 3, No. 1, 2021
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The transmission problem in linear isotropic elasticity

Plamen Stefanov, Gunther Uhlmann and András Vasy
Vol. 3 (2021), No. 1, 109–161
DOI: 10.2140/paa.2021.3.109
Abstract

We study the isotropic elastic wave equation in a bounded domain with boundary with coefficients having jumps at a nested set of interfaces satisfying the natural transmission conditions there. We analyze in detail the microlocal behavior of such solutions like reflection, transmission and mode conversion of S and P waves, evanescent modes, and Rayleigh and Stoneley waves. In particular, we recover Knott’s equations in this setting. We show that knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the P and the S waves if there is a strictly convex foliation with respect to them, under an additional condition of lack of full internal reflection of some of the waves.

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Keywords
elasticity, microlocal, inverse problem, transmission problem
Mathematical Subject Classification
Primary: 35A27, 35R30
Milestones
Received: 27 March 2020
Revised: 28 August 2020
Accepted: 2 October 2020
Published: 28 May 2021
Authors
Plamen Stefanov
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Gunther Uhlmann
Department of Mathematics
University of Washington
Seattle, WA
United States		 Department of Mathematics
University of Helsinki
Helsinki
Finland		 IAS, HKUST
Hong Kong
China
András Vasy
Department of Mathematics
Stanford University
Stanford, CA
United States