Borderline regularity in singular free boundary problems

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.