A duality approach to the fractional Laplacian with measure data

By Kenneth Hvistendahl Karlsen, Francesco Petitta, Suleyman Ulusoy. 2025

54 citations recorded by the source corpus.

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like ( ) s v = in R N ; with vanishing conditions at innity. Here is a bounded Radon measure whose support is compactly contained in R N , N 2, and () s is the fractional Laplace operator of order s2 (1=2; 1).

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bbaf12507aaf88e7a04d511f8609057b68f9df51

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

Hitchhiker's guide to the fractional Sobolev spaces (2011, background): “equations, are now experiencing impressive applications in different subjects, such as, among others, the thin obstacle problem [87, 69], optimization [37], finance [26], phase transitions [2, 14, 88, 40, 45], stratified materials [83, 23, 24], anomalous diffusion [68, 98, 65], crystal dislocation [”
Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian (2013, background): “[18] Kenneth H. Karlsen, Francesco Petitta, and Suleyman Ulusoy.”
Semilinear fractional elliptic equations involving measures (2013, methodology): “devoted to non-linear equations involving fractional Laplacian or more general integro-diﬀerential operators and we mention the reference [8, 9, 10, 14, 15, 24, 30, 32]. In particular, the authors in [23] used the duality approach to study the equations of (−∆)αv= µ in RN, where µis a Radon measure ”
On the fractional p-Laplacian equations with weight and general datum (2016, methodology): “with weight was considered in [5]. The authors proved the existence and the uniqueness of entropy solution for datum in L1. For the operator (−∆)s p,β, the case p= 2 and β= 0 was analyzed in [25] and [24]. Using a duality argument, in the sense of Stampacchia, the authors were able to prove the exis”
Nonlocal Equations with Measure Data (2014, background): “8) in the case s > 1/2 have been obtained in [29] and, in a different setting in [34].”
Dirichlet forms and semilinear elliptic equations with measure data (2012, background): “Solutions in the sense of duality of linear nonlocal elliptic equations with measure data are considered in [13] in case A = ∆α on Rd with α ∈ ((1) 2 , 1) and d ≥ 2.”