Borderline regularity in singular free boundary problems

José Miguel Urbano (KAUST and University of Coimbra)

Abstract: We address the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term. When the potential is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which is optimal in this general setting. In the one-phase case, however, we obtain gradient bounds along the free boundary, revealing a structural gain in regularity. Most notably, we prove that minimizers are continuously differentiable along the free boundary if the potential is continuous, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.