All Questions

Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e. $$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$ where $a$ are ...

Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...

I have a question on the regularity of a weak solution to an elliptic PDE with mixed boundary condition. Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ ...

Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$. Question: What is the Hausdorff dimension of the critical ...

Let $n\geq 3$, and $u$ satisfies $$ -\Delta u=K(x)u^{\frac{n+2}{n-2}} \quad x\in B_1\setminus \{0\}, $$ where $|K(x)|\leq A$ in $B_1\setminus \{0\}$, and $u\geq 0$ in $B_1\setminus \{0\}$. Can we ...

I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance, $$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...