On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions

Victor Isakov

doi:10.3934/ipi.2019044

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2019, Volume 13, Issue 5: 983-1006. Doi: 10.3934/ipi.2019044

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On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions

Victor Isakov

Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, USA

Received: September 2018

Revised: May 2019

Published: July 2019

The author is supported by NSF grant 15-14886 and the Emylou Keith and Betty Dutcher Distinguished Professorship

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Abstract

We derive bounds of solutions of the Cauchy problem for general elliptic partial differential equations of second order containing parameter (wave number) $ k $ which are getting nearly Lipschitz for large $ k $. Proofs use energy estimates combined with splitting solutions into low and high frequencies parts, an associated hyperbolic equation and the Fourier-Bros-Iagolnitzer transform to replace the hyperbolic equation with an elliptic equation without parameter $ k $. The results suggest a better resolution in prospecting by various (acoustic, electromagnetic, etc) stationary waves with higher wave numbers without any geometric assumptions on domains and observation sites.

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Mathematics Subject Classification: Primary: 35B60, 35R30, 35R35; Secondary: 65M32.

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