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On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions. (English) Zbl 1423.35447

Summary: 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.

MSC:

35R30 Inverse problems for PDEs
35B60 Continuation and prolongation of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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