Open AccessThe domain derivative for semilinear elliptic inverse obstacle problems
Author(s) -
Frank Hettlich
Publication year - 2022
Publication title -
inverse problems and imaging
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.755
H-Index - 40
eISSN - 1930-8345
pISSN - 1930-8337
DOI - 10.3934/ipi.2021071
Subject(s) - regularization (linguistics) , mathematics , inverse problem , domain (mathematical analysis) , boundary value problem , inverse , cauchy distribution , geodetic datum , boundary (topology) , mathematical analysis , derivative (finance) , nonlinear system , obstacle problem , obstacle , computer science , geometry , physics , cartography , quantum mechanics , political science , financial economics , law , economics , geography , artificial intelligence
We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.