Poisson Problems involving fractional Hardy operators and measures

Abstract

In this paper, we study the Poisson problem involving a fractional Hardy operator and a measure source. The complex interplay between the nonlocal nature of the operator, the peculiar effect of the singular potential and the measure source induces several new fundamental difficulties in comparison with the local case. To overcome these difficulties, we perform a careful analysis of the dual operator in the weighted distributional sense and establish fine properties of the associated function spaces, which in turn allow us to formulate the Poisson problem in an appropriate framework. In light of the close connection between the Poisson problem and its dual problem, we are able to establish various aspects of the theory for the Poisson problem including the solvability, a priori estimates, variants of Kato's inequality and regularity results.

Semilinear elliptic equations involving fractional Hardy operators

“…Further properties of L s µ were obtained in [5,25,29]. Recent years have witnessed a growing interest in the study of elliptic equations with fractional Hardy potentials, evidenced by [16,30,21,31,37,13,9]. Let us recall relevant results on the linear equation with a fractional Hardy potential.…”

“…In a recent paper [9], the authors of the present paper set up an appropriate distributional framework to investigate of the form…”

“…where ν is a bounded measure on Ω. The approach in [9] is to analyze the associated weighted fractional Laplace operator (−∆) s γ which is defined by…”

“…The results in [9], which will be recalled in Section §2, provide a basic framework for the study of problem (1.1) with Radon measures.…”

“…In this paper, we perform a deep analysis of this interplay and develop a theory for (1.1) in measure frameworks. The main ingredients includes recent results in [9] (see subsection 2.1), the local behavior near the origin of the solutions to the Poisson equation (see Proposition 2.5) obtained by adapting ideas in [15] and in [27], twosided estimates of approximating solutions expressed in term of the fundamental solution Φ Ω s,µ (see Lemma 5.1).…”