Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
Abstract
We construct smooth, uniformly convex planar domains that admit minimal, strictly stable solutions of a semilinear Dirichlet problem whose superlevel sets are nonetheless nonconvex. The class of admissible nonlinearities includes, in particular, two prototypical cases: the Gelfand-type nonlinearity eu and the family of shifted power-type nonlinearities (a+u)p, where a>0 and p>1. By applying the elementary scaling properties of the Dirichlet problem, we also show that the same lack of convexity of superlevel sets holds for the corresponding parameter-dependent equations. These results provide a negative answer to a question posed by Brezis, who inquired whether the stability of a solution necessarily entails quasiconcavity for these prototypical stable configurations.
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BibTeX
@misc{zhang2026strictly,
title = {{Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets}},
author = {Yi Ru-Ya Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.06031},
eprint = {2607.06031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.06031},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
AU - Yi Ru-Ya Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2607.06031
UR - https://arxiv.org/abs/2607.06031
AB - We construct smooth, uniformly convex planar domains that admit minimal, strictly stable solutions of a semilinear Dirichlet problem whose superlevel sets are nonetheless nonconvex. The class of admissible nonlinearities includes, in particular, two prototypical cases: the Gelfand-type nonlinearity e^u and the family of shifted power-type nonlinearities (a+u)^p, where a>0 and p>1. By applying the elementary scaling properties of the Dirichlet problem, we also show that the same lack of convexity of superlevel sets holds for the corresponding parameter-dependent equations. These results provide a negative answer to a question posed by Brezis, who inquired whether the stability of a solution necessarily entails quasiconcavity for these prototypical stable configurations.
ER -