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PreprintarXiv licencearXiv

An inverse free boundary problem

  • Cătălin I. Cârstea
  • Matti Lassas
  • Jinpeng Lu
  • Lauri Oksanen
  • Ziyao Zhao
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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.

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Cite

BibTeXRIS
BibTeX
@misc{carstea2026inverse,
  title = {{An inverse free boundary problem}},
  author = {Cătălin I. Cârstea and Matti Lassas and Jinpeng Lu and Lauri Oksanen and Ziyao Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.21153},
  eprint = {2608.21153},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.21153},
}
RIS
TY  - UNPB
TI  - An inverse free boundary problem
AU  - Cătălin I. Cârstea
AU  - Matti Lassas
AU  - Jinpeng Lu
AU  - Lauri Oksanen
AU  - Ziyao Zhao
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2608.21153
UR  - https://arxiv.org/abs/2608.21153
AB  - 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.
ER  -