Abstract

We consider the semilinear problem

$$\Delta u={\lambda }_{+}\left(-\hspace{0.17em}\text{log}\hspace{0.17em}{u}^{+}\right)1_{\{u>0\}}-{\lambda }_{-}\left(-\hspace{0.17em}\text{log}\hspace{0.17em}{u}^{-}\right)1_{\{u<0\}}\text{in}\mathrm{ }{B}_1,$$

where B₁ is the unit ball in ${\mathbb{R}}^n$ and assume ${\lambda }_{+},{\lambda }_{-}>0.$ Using a monotonicity formula argument, we prove an optimal regularity result for solutions: $\nabla u$ is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other (more serious) issue is a term in the Weiss energy which is potentially non-integrable unless one already knows the optimal regularity of the solution: this puts us in a catch-22 situation.

中文翻译：

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具有对数奇异性的两相类障碍问题的最优正则性

摘要

我们考虑半线性问题

$$\Delta 你={\lambda }_{+}\left(-\hspace{0.17em}\text{日志}\hspace{0.17em}{你}^{+}\right)1_{\{你>0\}}-{\lambda }_{-}\left(-\hspace{0.17em}\text{日志}\hspace{0.17em}{你}^{-}\right)1_{\{你<0\}}\text{在}\mathrm{ }{乙}_1,$$

其中B ₁是单位球${\mathbb{电阻}}^n$ 并假设 ${\lambda }_{+},{\lambda }_{-}>0.$ 使用单调性公式论证，我们证明了解决方案的最佳正则性结果： $\nabla 你$是一个 log-Lipschitz 函数。这个问题引入了两个主要困难。首先是问题的缩放和扩展缺乏不变性。另一个（更严重的）问题是 Weiss 能量中的一个项，它可能是不可积的，除非已经知道解决方案的最佳规律：这使我们处于 catch-22 情况。