# 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 $u_{tt} - Lu = 0 (0 < t ≤ T, x \in \Omega \subset \mathbb R^n)$ with linear densely defined self-adjoint and positive definite operators $L : D(L) \subset H \to H$ where $H$ denotes a Hilbert space with norm $\| \cdot \|$ and inner product $(\cdot, \cdot)$. We assume that instead of exact data $y = u(x,0)$ or $y = u_t(x,0)$ noisy data $y^{\delta} = u^{\delta} (x,0)$ or $y^{\delta} = u(x,0)$ are available, respectively, with $\|y – y^{\delta} \| ≤ \delta$. Furthermore we assume certain smoothness conditions $u(x,t) \in M$ 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 $\delta$. 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