Fractional elliptic equations, Caccioppoli estimates and regularity
Fractional elliptic equations, Caccioppoli estimates and regularity
Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\ u=0,&\hbox{on}~\partial\Omega, \end{cases}\quad \hbox{and}\quad \begin{cases} L^su=f,&\hbox{in}~\Omega,\\ \partial_Au=0,&\hbox{on}~\partial\Omega. \end{cases}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and $\partial_Au$ is the conormal derivative of $u$ with respect to …