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The obstacle problem for nonlinear integro-differential operators

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Abstract

We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator with measurable coefficients. Amongst other results, we will prove both the existence and uniqueness of the solutions to the obstacle problem, and that these solutions inherit regularity properties, such as boundedness, continuity and Hölder continuity (up to the boundary), from the obstacle.

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Notes

  1. For preliminary results in this direction, it is worth mentioning the very recent paper [6], where optimal regularity results of the solution to the obstacle problem, and of the free boundary near regular points, have been achieved for linear integro-differential operators as in (1) in the case when \(p=2\).

  2. When needed, our definition of Tail can also be given in a more general way by replacing the ball \(B_r\) and the corresponding \(r^{sp}\) term by an open bounded set \(E\subset {\mathbb {R}}^n\) and its rescaled measure \(|E|^{sp/n}\), respectively. This is not the case in the present paper.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 1100, 00076, Aalto, Finland

    Janne Korvenpää & Tuomo Kuusi

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Campus-Parco Area delle Scienze, 53/A, 43124, Parma, Italy

    Giampiero Palatucci

Authors
  1. Janne Korvenpää
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  2. Tuomo Kuusi
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  3. Giampiero Palatucci
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Corresponding author

Correspondence to Giampiero Palatucci.

Additional information

Communicated by A. Malchiodi.

The first author has been supported by the Magnus Ehrnrooth Foundation (Grant Nos. ma2014n1, ma2015n3). The second author has been supported by the Academy of Finland. The third author is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica “F. Severi” (INdAM), whose support is acknowledged. The results of this paper have been announced in the preliminary research report [22].

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Korvenpää, J., Kuusi, T. & Palatucci, G. The obstacle problem for nonlinear integro-differential operators. Calc. Var. 55, 63 (2016). https://doi.org/10.1007/s00526-016-0999-2

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