Identification of a discontinuous source in the heat equation. (English) Zbl 0986.35129

From the introduction: Let \(D\) be a domain contained in a domain \(\Omega\subseteq \mathbb{R}^2\). We consider the initial boundary value problem \[ \partial_t u-\Delta u=\chi_D \quad\text{in }\Omega \times(0,T) \] with \(u(x,0)=0\), \(x\in\Omega\) and \(u(x,t)=0\) for \((x,t)\in\partial \Omega\times(0,T)\) in a finite time interval \([0,T]\). The purpose of this paper is to investigate the feasibility of recovering the domain \(D\) from information on the flux, that is, the normal derivative of \(u\), on the exterior boundary \(\partial\Omega\).
It is shown that a minimal set of data according to uniqueness of the inverse problem is given by flux measurements in time at two distinct points on the boundary. Besides the uniqueness result iterative regularization schemes are developed. The methods are based on the domain derivative and a general existence theory as well as a representation of the domain derivative for parabolic equations is derived.


35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
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