Methods of the Perturbation Theory for Fundamental Solutions to the Generalization of the Fractional Laplaciane

Abstract

We study the regularity properties of the solutions to the fractional Laplacian equation with perturbations. The Harnack inequality of a weak solution u∈WspRl to the fractional Laplacian problem is established, and the oscillation of the solution to the fractional Laplacian is estimated. We show that let 1≤p<∞ and s∈01, and let u∈Wsp be a weak solution to Lu=0inΩ, with the condition u=finRl\\Ω, where function f belongs to the Sobolev space WspRl. Then, the function u∈Wsp is locally Holder continuous and oscillation of the function satisfies the estimation oscBx0ru≤Cδspp−1upBx0ρp+ +Cδspp−1maxu0p−11⋅−x0l+spRl\\Bx0ρ1p−1 holds for δ∈01 and for all r,ρ such that 0<r<ρ. Also, let u∈WspRl be a weak solution to the boundary problem for the fractional Laplacian −Δps−b⋅∇u=0inΩ, and on the boundary u=finRl\\Ω, where 1≤p<∞ and s∈01, and let u be nonnegative in a ball function with the center x0 with the radius ρ, then the following estimation supBx0ru≤CinfBx0ru+Crρspp−1max−u0p−11⋅−x0l+spRl\\Bx0ρ1p−1 holds for r,ρ such that 0<r<ρ.