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Well posedness for the Poisson problem on closed Lipschitz manifolds

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Abstract

We study the weak formulation of the Poisson problem on closed Lipschitz manifolds. Lipschitz manifolds do not admit tangent spaces everywhere and the definition of the Laplace–Beltrami operator is more technical than on classical differentiable manifolds (see, e.g., Gesztesy in J Math Sci 172:279–346, 2011). They however arise naturally after the triangulation of a smooth surface for computer vision or simulation purposes. We derive Stokes’ and Green’s theorems as well as a Poincaré’s inequality on Lipschitz manifolds. The existence and uniqueness of weak solutions of the Poisson problem are given in this new framework for both the continuous and discrete problems. As an example of application, numerical results are given for the Poisson problem on the boundary of the unit cube.

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Authors and Affiliations

  1. CEA-Saclay, Université Paris-Saclay, DES, ISAS, DM2S, STMF, 91191, Gif-sur-Yvette, France

    Michaël Ndjinga

  2. Department of Mathematics, Faculty of Sciences, University of Yaounde 1, P.O. Box 812, Yaoundé, Cameroon

    Marcial Nguemfouo

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  1. Michaël Ndjinga
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  2. Marcial Nguemfouo
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Correspondence to Marcial Nguemfouo.

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This article is part of the section “Computational Approaches” edited by Siddhartha Mishra.

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Ndjinga, M., Nguemfouo, M. Well posedness for the Poisson problem on closed Lipschitz manifolds. Partial Differ. Equ. Appl. 4, 44 (2023). https://doi.org/10.1007/s42985-023-00263-x

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  • DOI: https://doi.org/10.1007/s42985-023-00263-x

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