The Dirichlet elliptic problem involving regional fractional Laplacian

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha_\Omega u=f \qquad & {\rm in}\quad \Omega,\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=g\quad & {\rm on}\quad \partial \Omega, \end{array} \end{equation} where $\mathrm{\Omega }$ is a bounded open domain in ${\mathbb{R}}^N$ ($N\ge 2$)with$C^2$boundary$\partial\Omega$,$\alpha\in(\frac12,1)$andtheoperator$(-\Delta)^\alpha_\Omega$ denotes the regional fractional Laplacian. We prove that when $g\equiv 0$, problem (\ref{0}) admits a unique weak solution in the cases that $f\in L^2(\mathrm{\Omega })$, $f\in L^1(\Omega, \rho^\beta dx)$and$f\in $\mathcal{M}$(\Omega,\rho^\beta)$,here$\rho(x)={\rm dist}(x,\partial\Omega)$,$\beta=2\alpha-1$and$$\mathcal{M}$(\Omega,\rho^\beta)$isaspaceofallRadonmeasures$\nu$satisfying$\int_\Omega \rho^\beta d|\nu|<+\infty.$ Finally, we provide an Integral by Parts Formula for the classical solution of (\ref{0}) with general boundary data $g$.