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Title: Weighted Sobolev space theory for non-local elliptic and parabolic equations with nonzero exterior condition on C1,1 open sets

Abstract: We introduce a weighted Sobolev space theory for the non-local elliptic equation Delta(alpha/2)u=f, x is an element of O; r O(over bar)cu=g as well as for the non-local parabolic equation u(t)=Delta(alpha/2)u+f,t>0,x is an element of O;rOu(0,& sdot;)=u0,r((0,T)xO(over bar))cu=g. Here alpha is an element of(0,2) and O is a C(1,1 )open set. We prove uniqueness and existence results in weighted Sobolev spaces. We measure the Sobolev and H & ouml;lder 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 O are less affected by those of exterior conditions on O(over bar)c. For instance, even if g=delta x0, the Dirac delta distribution concentrated at x0 is an element of O(over bar)c, the solution to the elliptic equation given with f=0 is infinitely differentiable in O, and for any k=0,1,2,3,& ctdot;, epsilon>0, and delta is an element of(0,1), it holds that |d(x)(-alpha 2+epsilon+k)D(x)(k)u(|Cb(O))+|dx(-alpha/2+epsilon+k+delta)D(x)(k)u|(C delta(O))= 2.

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