Abstract
We study bounded trace maps on hypersurfaces for Sobolev spaces from a point of view that is fundamentally different from the one in the classical theory. This allows us to construct bounded trace maps under weak regularity assumptions on the hypersurfaces. In the case of bounded domains in \(\mathbf{R}^n\) we only require the continuity of the boundary. For hypersurfaces in the whole space \(\mathbf{R}^n\) we only assume that the hypersurfaces are Lebesgue measurable. As an application of our trace maps we consider the Dirichlet problem and we prove a coarea formula where the level sets are only assumed to be Lebesgue measurable hypersurfaces.
In Memory of Erik Balslev
Research partially supported by project PAPIIT-DGAPA UNAM IN103918 and by project SEP-CONACYT CB 2015, 254062. R. Weder—Fellow, Sistema Nacional de Investigadores.
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Acknowledgements
This paper was partially written while I was visiting INRIA Saclay Île-de-France and ENSTA. I thank Anne-Sophie Bonnet-BenDhia and Patrick Joly for their kind hospitality. I thank Vladimir Maz’ya for his detailed information on the literature on trace maps and on the coarea formula.
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Weder, R. (2021). Trace Maps Under Weak Regularity Assumptions. In: Albeverio, S., Balslev, A., Weder, R. (eds) Schrödinger Operators, Spectral Analysis and Number Theory. Springer Proceedings in Mathematics & Statistics, vol 348. Springer, Cham. https://doi.org/10.1007/978-3-030-68490-7_14
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DOI: https://doi.org/10.1007/978-3-030-68490-7_14
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