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- ItemUniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Hömberg, Dietmar; Lu, Shuai; Yamamoto, MasahiroWe consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.
- ItemOverdetermined problems for the fractional Laplacian in exterior and annular sets(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Soave, Nicola; Valdinoci, EnricoWe consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.