Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian

Abstract

For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive the $L^p$-estimate for fractional Laplacian operators to better understand this phenomena. Several weighted fractional Sobolev spaces appear naturally. We then establish the embedding relations between these spaces. These existence-uniqueness conditions and the spaces we introduce here are intrinsically related to the fractional Laplacian. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems.

References in corpus (5)

• An extension problem related to the fractional Laplacian
• Regularity theory for fully nonlinear integro-differential equations
• Some observations on the Green function for the ball in the fractional Laplace framework
• On the maximum principle for higher-order fractional Laplacians
• Existence theorems of the fractional Yamabe problem

Cited by in corpus (1)

• On the Dirichlet problem for fractional Laplace equation on a general domain