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Algebra properties for Sobolev spaces — applications to semilinear PDEs on manifolds

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Journal d'Analyse Mathématique Aims and scope

Abstract

In this work, we aim to prove algebra properties for generalized Sobolev spaces W s,p ∩ L ∞ on a Riemannian manifold (or more general homogeneous type space as graphs), where W s,p is of Bessel-type W s,p:= (1+L)−s/m(L p) with an operator L generating a heat semigroup satisfying off-diagonal decays. We do not require any assumption on the gradient of the semigroup. Instead, we propose two different approaches (one by paraproducts associated to the heat semigroup and another one using functionals). We also study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non-bounded geometry) and even in euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.

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Authors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon, 43 Boulevard Du 11 Novembre 1918, F-69622, Villeurbanne Cedex, France

    Nadine Badr

  2. CNRS — Université de Nantes, Laboratoire Jean Leray, 2, Rue de la Houssinière, 44322, Nantes Cedex 3, France

    Frédéric Bernicot

  3. Institut Fourier, Université Joseph Fourier Grenoble I, 100 Rue des Maths, BP 74, 38402, Saint-Martin-D’Hères Cedex, France

    Emmanuel Russ

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  1. Nadine Badr
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  2. Frédéric Bernicot
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  3. Emmanuel Russ
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Correspondence to Nadine Badr.

Additional information

First author is supported by the ANR under the project AFoMEN no. 2011-JS01-001-01.

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Badr, N., Bernicot, F. & Russ, E. Algebra properties for Sobolev spaces — applications to semilinear PDEs on manifolds. JAMA 118, 509–544 (2012). https://doi.org/10.1007/s11854-012-0043-1

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  • DOI: https://doi.org/10.1007/s11854-012-0043-1

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