A Hölder 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 $$Lu -\lambda u = f \quad \text{in} \quad \mathbb{R}^{d}, $$ where $$Lu(x)={\int}_{\mathbb{R}^{d}}(u(x+y)-u(x)-y \cdot \nabla u(x) \chi (y) )\, a(y)J(y)dy. $$ Here χ(y) is a suitable indicator function, J(y)d y is a rotationally invariant Levy measure on ℝ d (i.e. \({\int }_{\mathbb {R}^{d}}\left (1\wedge |y|^{2}\right )J(y)dy<\infty \)), and a(y) is an only measurable function with positive lower and upper bounds.