Year: 2026
Type: preprint
Abstract: We study inverse problems for the elliptic and parabolic obstacle problems from boundary measurements. For the classical elliptic obstacle problem with strictly superharmonic obstacle function, we show that the Dirichlet-to-Neumann map admits a one-sided linearization at every boundary datum lying strictly above the obstacle. The linearized map is the Dirichlet-to-Neumann map for a rough Dirichlet problem on the a priori unknown non-contact set. We show that these linearized Cauchy data uniquely determine the non-contact set up to set of Sobolev $2$-capacity zero and consequently determine the obstacle. Our result applies to the inverse problem for a parabolic obstacle problem where both the coefficient and the obstacle function are time-independent by reducing to the elliptic inverse problem.
Source: arXiv (Cornell University)
Institution: University of Helsinki
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6218650241, G8144583337
OpenAlex ID: W7204092785