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The Dirichlet-to-Neumann operator associated with the 1-Laplacian and evolution problems

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Abstract

In this paper, we present first insights about the Dirichlet-to-Neumann operator in \(L^{1}\) associated with the 1-Laplace operator or total variational flow operator. This operator is the main object, for example, in studying inverse problems related to image processing, but also admits an important relation to geometry. We show that this operator can be represented by the sub-differential in \(L^1\times L^{\infty }\) of a convex, homogeneous, and continuous functional on \(L^{1}\). This is quite surprising since it implies a type of stability or compactness result even though the singular Dirichlet problem governed by the 1-Laplace operator might have infinitely many weak solutions (if the given boundary data is not continuous). As an application, we obtain well-posedness and long-time stability of solutions of a singular coupled elliptic-parabolic initial boundary-value problem.

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Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, 2006, Australia

    Daniel Hauer

  2. Departamento de Análisis Matemático, Universitat de València, Valencia, Spain

    José M. Mazón

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  1. Daniel Hauer
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  2. José M. Mazón
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Correspondence to Daniel Hauer.

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Communicated by Y. Giga.

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Daniel Hauer is very grateful for the kind invitation to the Universitat de València and their hospitality. In the second half of the research project, he was supported by the Australian Research Council Grant DP200101065. José M. Mazón was partially supported by the Spanish MCIU and FEDER, Project PGC2018-094775-B-100.

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Hauer, D., Mazón, J.M. The Dirichlet-to-Neumann operator associated with the 1-Laplacian and evolution problems. Calc. Var. 61, 37 (2022). https://doi.org/10.1007/s00526-021-02149-5

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