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

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

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  • 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.

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


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