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Wavelets and regularization of the Cauchy problem for the Laplace equation. (English) Zbl 1135.35093

Summary: The Cauchy problem for the two-dimensional Laplace equation in the strip \(0<x \leqslant \)1 is considered. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data; a small perturbation in the data can cause a dramatically large error in the solution for \(0<x\leqslant \)1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in \(H^r(\mathbb R)\)-norm is also provided.

MSC:

35R25 Ill-posed problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B35 Stability in context of PDEs
65T60 Numerical methods for wavelets
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References:

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