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Sobolev spaces and \(\nabla \)-differential operators on manifolds I: basic properties and weighted spaces

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Abstract

We study covariant Sobolev spaces and \(\nabla \) -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted \(\nabla \)). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the \(\nabla \)-differential operators on general manifolds. For instance, we prove map** properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection \(\nabla \), as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our \(\nabla \)-differential operators. We also introduce and study the notion of a \(\nabla \)-bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for \(\nabla \)-differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

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Acknowledgements

We thank Herbert Amann, Bernd Ammann, Nadine Große, Sergiu Moroianu, and Radu Purice for useful discussions. We thank Camille Laurent-Gengoux and Martin Bordemann for useful references. We also thank the referees for carefully reading the paper. Our results do not require any supporting data.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania

    Mirela Kohr

  2. UFR MIM, Ile du Saulcy, CS 50128, Université de Lorraine, 57045, Metz, France

    Victor Nistor

  3. Institute of Mathematics of the Romanian Academy, PO BOX 1-764, 014700, Bucharest, Romania

    Victor Nistor

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  1. Mirela Kohr
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Correspondence to Mirela Kohr.

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In memory of Professor Gabriela Kohr, with deep respect.

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M.K. has been partially supported by AGC35124/31.10.2018. V.N. has been partially supported by ANR-14-CE25-0012-01.

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Kohr, M., Nistor, V. Sobolev spaces and \(\nabla \)-differential operators on manifolds I: basic properties and weighted spaces. Ann Glob Anal Geom 61, 721–758 (2022). https://doi.org/10.1007/s10455-022-09824-6

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