Lipschitz stability in inverse parabolic problems by the Carleman estimate. (English) Zbl 0992.35110

Summary: We consider a system \(y_t(t,x)=-\mathcal Ay(t,x)+g(t,x)\) \((0<t<T\), \(x\in\Omega\)), \(y(\theta,x)=y_0(x)\) \((x\in \Omega)\) with a suitable boundary condition, where \(\Omega\subset\mathbb{R}^n\) is a bounded domain, \(-\mathcal A\) is a uniformly elliptic operator of second order whose coefficients are suitably regular for \((t,x)\), \(\theta\in ]0,T[\) is fixed, and the function \(g(t,x)\) satisfies \(|g_t(t,x)|\leq C|g(\theta,x)|\) for \((t,x)\in [0,T]\times\overline{\Omega}\). Our inverse problems are determinations of \(g\) using overdetermining data \(y|_{]0,T[\times\omega}\) or \(\{y|_{]0,T[\times\Gamma_0}\), \(\nabla y|_{]0,T[\times\Gamma_0}\}\), where \(\omega\subset\Omega\) and \(\Gamma_0\subset\partial\Omega\). Our main result is the Lipschitz stability in these stability problems. We also consider the determination of \(f=f(x)\), \(x\in \Omega\) in the case of \(g(t,x)=f(x)R(t,x)\) with given \(R\) satisfying \(R(\theta,\cdot)>0\) on \(\overline\Omega\). Finally, we discuss an upper estimation of our overdetermining data by means of \(f\).


35R30 Inverse problems for PDEs
35R35 Free boundary problems for PDEs
35K10 Second-order parabolic equations
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