All Questions

For $s\in(0,1],$ consider the following non-local fractional laplacian: $$(-\Delta)^sv= f ~~\text{on } \mathbb{R}^n.$$ Then how to use "the standard elliptic estimate" to obtain: for $p\in[...

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...

Let $\{ \phi _{d,m}\}_{m\geq 1}$ be multi-dimensional spherical harmonics, i.e., solutions of $\Delta \phi = E\phi$ on the sphere $S^d$ for $d>1$, arranged in an increasing order $E_1 \leq E_2 \leq ...

Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator $$ Lu:=-\Delta u+V(x)u $$ where $V\geq0$ is the electric potential, ...

Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...

I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...