A Holder Regularity Theory for a Class of Non-Local Elliptic Equations Related to Subordinate Brownian Motions

Abstract

In this article we present the existence, uniqueness, and Holder regularity of solutions for the non-local elliptic equations of the type Lu-lambda u = f in R-d, where Lu(x) = integral(Rd) (u(x + y) - u(x) - y.del u(x)chi(y)) a(y) J (y) dy. Here chi(y) is a suitable indicator function, J(y) dy is a rotationally invariant Levy measure on R-d (i. e. integral(Rd) (1 boolean AND vertical bar y vertical bar(2) )J(y) dy < infinity), and a(y) is an only measurable function with positive lower and upper bounds.