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Nonlocal Equations with Measure Data

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Abstract

We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149–169, 1989, Partial Differ Equ 17:641–655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591–613, 1992, Acta Math 172:137–161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón–Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321–381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.

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Authors and Affiliations

  1. Aalto University Institute of Mathematics, P.O. Box 11100, 00076, Aalto, Finland

    Tuomo Kuusi

  2. Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, Campus, 43100, Parma, Italy

    Giuseppe Mingione

  3. Université Aix-Marseille, I2M, Technopole de Chateaux-Gombert, Marseille, France

    Yannick Sire

Authors
  1. Tuomo Kuusi
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  2. Giuseppe Mingione
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Correspondence to Yannick Sire.

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Communicated by L. Caffarelli

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Kuusi, T., Mingione, G. & Sire, Y. Nonlocal Equations with Measure Data. Commun. Math. Phys. 337, 1317–1368 (2015). https://doi.org/10.1007/s00220-015-2356-2

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