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Nonlocal self-improving
properties
Tuomo Kuusi, Giuseppe Mingione and Yannick Sire |
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Vol. 8 (2015), No. 1, 57–114
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Abstract |
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Solutions to nonlocal equations with measurable coefficients are higher differentiable. Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by where the kernel is a measurable function and satisfies the bounds with , , while for some . The main result states that there exists a positive, universal exponent such that for every weak solution the self-improving property holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Hölder-type inequalities in and which is implied by delicate covering and exit-time arguments. In turn, such reverse Hölder inequalities are based on the concept of dual pairs, that is, pairs of measures and functions in which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order . |
Keywords
elliptic equations, fractional differentiability, nonlocal
operators
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Mathematical Subject Classification 2010
Primary: 35D10
Secondary: 35R11
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Milestones
Received: 19 February 2014
Accepted: 22 October 2014
Published: 15 April 2015
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Authors |
| Tuomo Kuusi | |
| Institute of Mathematics Aalto University P.O. Box 11100 FI-00076 Aalto Finland |
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| Giuseppe Mingione | |
| Dipartimento di Matematica e
Informatica Universitá di Parma Parco Area delle Scienze 53/a, Campus I-43100 Parma Italy |
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| Yannick Sire | |
| Institut de Mathématique de
Marseille CMI-UMR CNRS 7353 Université Aix-Marseille 9 rue F. Joliot Curie 13453 Marseille Cedex 13 France |
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