Uniqueness of one-phase cones with isolated singularity

@misc{carducci2026uniqueness,
  title = {{Uniqueness of one-phase cones with isolated singularity}},
  author = {Matteo Carducci and Bozhidar Velichkov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.35175},
  eprint = {2609.35175},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35175},
}

TY  - UNPB
TI  - Uniqueness of one-phase cones with isolated singularity
AU  - Matteo Carducci
AU  - Bozhidar Velichkov
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.35175
UR  - https://arxiv.org/abs/2609.35175
AB  - We prove uniqueness of the blow-up at every singular point of the one-phase free boundary problem for which one blow-up has an isolated singularity. The result applies to Lipschitz stationary solutions and does not require any integrability assumption on the cone. In this sense, our result completes the picture for uniqueness of tangent cones at isolated singularities in the one-phase problem. Inspired by Simon's work in the minimal surface setting, we prove an infinite dimensional Łojasiewicz inequality for the spherical Weiss' energy. This yields an epiperimetric inequality for non-minimizing solutions, which leads to the uniqueness result.
ER  -