| Abstract: |
| We address the borderline regularity of local minimizers of singular energy functionals. For bounded and measurable potentials, we show that sign-changing minimizers are Log-Lipschitz continuous, which is optimal in this generality. In the one-phase case, however, we derive gradient bounds along the free boundary, uncovering a structural gain in regularity. Our first main result establishes sharp Lipschitz regularity for a merely bounded potential. Most notably, we prove that if the potential is further assumed to be a modulus of continuity, then minimizers become continuously differentiable. We thus identify a sharp threshold for differentiability in terms of the potential`s regularity. |
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