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Gradient bounds for the horizontal p-Laplacian on a Carnot group and some applications

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Abstract

In a Carnot group we prove a new priori bound for the right-invariant horizontal gradient of smooth solutions of a class of quasilinear equations which are modeled on the so-called horizontal p-Laplacian. Exploiting such bound and a regularization procedure based on difference quotients we obtain the \({C^{1,\alpha}_{loc}}\) regularity of weak solutions which possess some special symmetries. For instance, in the first Heisenberg group \({\mathbb{H}^{1}}\) we obtain such regularity for all weak solutions of the horizontal p-Laplacian, with p ≥ 2, which are of the form u(z, t) = u(|z|, t).

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Nicola Garofalo

  2. Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Padova, 35131, Padova, Italy

    Nicola Garofalo

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  1. Nicola Garofalo
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Correspondence to Nicola Garofalo.

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Supported in part by NSF Grant DMS-0701001

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Garofalo, N. Gradient bounds for the horizontal p-Laplacian on a Carnot group and some applications. manuscripta math. 130, 375–385 (2009). https://doi.org/10.1007/s00229-009-0294-z

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  • DOI: https://doi.org/10.1007/s00229-009-0294-z

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