Weighted Sobolev space theory for non-local elliptic and parabolic equations with non-zero exterior condition on $C^{1,1}$ open sets

Abstract

We introduce a weighted Sobolev space theory for the non-local elliptic equation $$ \Delta^{\alpha/2}u=f, \quad x\in \mathcal{O}\,; \quad r_{\overline{\mathcal{O}}^c}u=g $$ as well as for the non-local parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in \mathcal{O} \,; \quad r_{\mathcal{O}}u(0,\cdot)=u_0, \,r_{(0,T)\times \overline{\mathcal{O}}^c}u=g. $$ Here, $\alpha\in (0,2)$ and $\mathcal{O}$ is a $C^{1,1}$ open set. We prove uniqueness and existence results in weighted Sobolev spaces. We measure the Sobolev and H\"older regularities of arbitrary order derivatives of solutions using a system of weights consisting of appropriate powers of the distance to the boundary. One of the most interesting features of our results is that, unlike the classical results in Sobolev spaces without weights, the weighted regularities of solutions in $\mathcal{O}$ are less affected by those of exterior conditions on $\overline{\mathcal{O}}^c$. For instance, even if $g=\delta_{x_0}$, the dirac delta distribution concentrated at $x_0\in \overline{\mathcal{O}}^c $, the solution to the elliptic equation given with $f=0$ is infinitely differentiable in $\mathcal{O}$, and for any $k=0,1,2, 3,\cdots$, $\varepsilon>0$, and $\delta\in (0,1)$, it holds that $$ |d_x^{-\frac{\alpha}{2}+\varepsilon+k}D^k_xu|_{C_b(\mathcal{O})} +|d_x^{-\frac{\alpha}{2}+\varepsilon+k+\delta} D^k_xu|_{C^{\delta}(\mathcal{O})}<\infty, $$ where $d_x=dist(x, \partial \mathcal{O})$.