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Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define $$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$ Is it true that $V(z) \in C^{\infty}(\partial \Omega)$? Mo …

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to \begin{equation} \begin{cases} -\Delta u=0 \quad &\mbox{in $\Omega$}\\ \frac{\partial u}{\ …

Let $u$ be a solution to the equation $$\Delta u=f(u,\nabla u)$$ where $f>0$ is a smooth function with $f f_{uu} \le 2f_u^2$. The seminal constant rank theorem states that if $D^2 u$ is positive semid …

I am motivated by the following question: Given a uniform heat source in a convex domain, and suppose that the outside temperature is equal to $0$, can we determine where the long-time temperature in …

Let $L=\Delta + c$ in 3 dimensions, where $c$ is a positive constant. I met this modified mean value property of a solution $u$ of $Lu=0$ as $$u(\xi)=\frac{\sqrt{c}\rho}{sin(\sqrt{c}\rho)}\frac{1}{4 …

I'm a beginner of the area of free boundary problem. Let me first give some background: $\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph. C …

I'm reading the first chapter of the book A geometric approach to free boundary problems by Caffarelli and Salsa, see the PDF here. The question came from Page 14—15. Let me state my question: Let $f …

Recall the proof of the $(\Lambda,r_0)$-perimeter minimizer. We say that $E$ is a $(\Lambda,r_0)$-perimeter minimizer in an open set A provided $sptD\chi_E=\partial E$ and there exist two constants …