A duality approach to the fractional Laplacian with measure data. (English) Zbl 1208.35162

Karlsen, Kenneth H.; Petitta, Francesco; Ulusoy, Suleyman

Publ. Mat., Barc. 55, No. 1, 151-161 (2011).

Summary: We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like
\[ (-\Delta)^sv=\mu\quad\text{in }\mathbb R^N, \]
with vanishing conditions at infinity. Here \(\mu\) is a bounded Radon measure whose support is compactly contained in \(\mathbb R^N\), \(N\geq 2\), and \(-(\Delta)^s\) is the fractional Laplace operator of order \(s\in(1/2,1)\).

MSC:

35R11	Fractional partial differential equations
35B40	Asymptotic behavior of solutions to PDEs
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Keywords:

fractional Laplacian; measure data; existence; uniqueness; duality solutions