标题： 奇异自由边界问题中的临界正则性 显示英文标题

标题： Borderline regularity in singular free boundary problems

摘要： 在本文中，我们研究在势能项$\sigma$的最小假设下，能量泛函局部极小值的边界正则性。当$\sigma$仅为有界可测时，我们证明变号极小值是 Log-Lipschitz 连续的，这在这一一般设置中代表了最优正则性。然而，在单相情况下，我们建立了极小值在其自由边界上的梯度估计，揭示了正则性中的结构优势。最显著的是，我们证明如果$\sigma$是连续的，那么极小值在自由边界上属于$C^1$类，从而根据势能的正则性确定了可微性的尖锐阈值。 显示英文摘要

摘要： In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term $\sigma$. When $\sigma$ 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 $\sigma$ is continuous, then minimizers are of class $C^1$ along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.