Borderline regularity in singular free boundary problems
Abstract
In this article, we investigate the borderline regularity of local minimizers of energy functionals of the form ∫12|Du|2+σ(u),under minimal assumptions on the potential term σ. When σ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if σ is continuous, then minimizers are of class C1 along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.
Fields of Study & Subjects
Differential Equations and Boundary Problems
Spectral Theory in Mathematical Physics
Nonlinear Differential Equations Analysis
Publication details
- Journal / source
- Communications in Partial Differential Equations
- Publisher
- Informa UK Limited
- Volume
- 51
- Issue
- 6
- Pages
- 619–636
- Publication date
- 3 June 2026
- Language
- EN