A stability estimate for a Cauchy problem for an elliptic partial differential equation

Abstract

A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y = 0, and boundary data are for x = 0 and x = 1. The elliptic operator is self-adjoint with nonconstant, smooth coefficients. The solution for y = 1 is sought. This Cauchy problem is ill-posed in an L 2 -setting. A stability functional is defined, for which a differential inequality is derived. Using this inequality a stability result of Hölder type is proved. It is demonstrated explicitly how the stability depends on the smoothness of the coefficients. The results can also be used for rectangle-like regions that can be mapped conformally onto a rectangle.

On the Cauchy problem for semilinear elliptic equations

“…As shown in [18], the solution u ∈ C([0, M]; H) is a weak solution of (1.1) if u satisfies the integral equation 5) where ϕ n = ϕ, φ n and f n (u)(s) = f (s, u(s)), φ n ) . Since z > 0 , we know from (2.5) that, when n becomes large, the terms cosh √ λ n z and sinh √ λ n (z − s) increase rather quickly.…”

“…(1.1), see e.g. [3,4,5,6,7,9,10,13,14,16,17] to mention only a few. On the other hand, the Cauchy problem for nonlinear elliptic equations has been much less investigated, [11,21], and it is the purpose of this study to make advances into the semi-linear problem (1.1).…”

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

“…For instance, Elden and Berntsson [14] used the logarithmic convexity method to obtain a stability result of Hölder type. Alessandrini et al [1] provided optimal stability results under minimal assumptions, whilst Reginska and Tautenhahn [32] presented some stability estimates and a regularization method for a Cauchy problem for Helmholtz equation.…”

Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

“…The Cauchy problem for an elliptic equation is a classical ill-posed problem and occurs in several important applications, such as inverse scattering [17,37], electrical impedance tomography [10], optical tomography [7], and thermal engineering [23]. The topic is treated in several monographs [16,36,37,41,42,45], and in numerous papers, see [1,3,6,8,9,18,24,25,34,47,48,51,57,61] and the references therein. Even if some of the theoretical investigations are quite general, the numerical procedures proposed are typically for the two dimensional case and often only valid for the problem with constant coefficients.…”