论文标题
对于对数奇异性的两阶段障碍物问题的最佳规律性
Optimal regularity for a two-phase obstacle-like problem with logarithmic singularity
论文作者
论文摘要
我们考虑半连续问题\ [ΔU=λ_+ \左（ - \ log u^u^+ \右）1 _ {\ {\ {U> 0> 0 \}} - λ_-\ lest（ - \ log u^ - \ u^ - \ right） $ \ mathbb {r}^n $中的单位球，假设$λ_+，λ_-> 0 $。使用单调性公式参数，我们证明了解决方案的最佳规律性结果：$ \ nabla u $是log-lipschitz函数。 这个问题引入了两个主要困难。首先是问题的扩展和爆炸缺乏不变性。另一个（更严重的）问题是Weiss Energy中的一个术语，除非有人已经知道解决方案的最佳规律性，否则它可能是不可集中的：这使我们处于捕获22的情况下。
We consider the semilinear problem \[ Δu = λ_+ \left(-\log u^+\right) 1_{\{u > 0\}} - λ_- \left(-\log u^- \right) 1_{\{u < 0\}} \qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and assume $λ_+, λ_- > 0$. 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.
