# Optimal Stable Solution of Cauchy Problems for Elliptic Equations

### Ulrich Tautenhahn

University of Applied Sciences, Zittau, Germany

## Abstract

We consider ill-posed Cauchy problems for elliptic partial differential equations $u_{tt}−Lu=0(0<t≤T,x∈Ω⊂R_{n})$ with linear densely defined self-adjoint and positive definite operators $L:D(L)⊂H→H$ where $H$ denotes a Hilbert space with norm $∥⋅∥$ and inner product $(⋅,⋅)$. We assume that instead of exact data $y=u(x,0)$ or $y=u_{t}(x,0)$ noisy data $y_{δ}=u_{δ}(x,0)$ or $y_{δ}=u(x,0)$ are available, respectively, with $∥y–y_{δ}∥≤δ$. Furthermore we assume certain smoothness conditions $u(x,t)∈M$ with appropriate sets $M$ and answer the question concerning the best possible accuracy for identifying $u(x,t)$ from the noisy data. For special sets $M$ the best possible accuracy depends either in a Hölder continuous way or in a logarithmic way on the noise level $δ$. Furthermore, we discuss special regularization methods which realize this best possible accuracy.

## Cite this article

Ulrich Tautenhahn, Optimal Stable Solution of Cauchy Problems for Elliptic Equations. Z. Anal. Anwend. 15 (1996), no. 4, pp. 961–984

DOI 10.4171/ZAA/740