Abstract
This paper is devoted to analyze the Dirichlet problem for a nonlinear elliptic equation involving the 1-Laplacian and a total variation term, that is, the inhomogeneous case of the equation arising in the level set formulation of the inverse mean curvature flow. We study this problem in an open bounded set with Lipschitz boundary. We prove an existence result and a comparison principle for nonnegative \(L^1\)-data. Moreover, we search the summability that the solution reaches when more regular \(L^p\)-data, with \(1<p<N\), are considered and we give evidence that this summability is optimal. To prove these results, we apply the theory of \(L^\infty \)-divergence measure fields which goes back to Anzellotti (Ann Mat Pura Appl (4) 135:293–318, 1983). The main difficulties of the proofs come from the absence of a definition for the pairing of a general \(L^\infty \)-divergence measure field and the gradient of an unbounded BV-function.
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Acknowledgements
This research has been partially supported by the Spanish Ministerio de Economía y Competitividad and FEDER, under project MTM2015–70227–P. The first author was also supported by Ministerio de Economía y Competitividad under Grant BES–2013–066655. The authors would like to thank Salvador Moll for some fruitful discussions concerning this paper.
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Latorre, M., Segura de León, S. Existence and comparison results for an elliptic equation involving the 1-Laplacian and \(L^1\)-data. J. Evol. Equ. 18, 1–28 (2018). https://doi.org/10.1007/s00028-017-0388-0
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DOI: https://doi.org/10.1007/s00028-017-0388-0
Keywords
- Nonlinear elliptic equations
- \(L^1\)-data
- 1-Laplacian operator
- Total variation term
- Comparison principle
- Inverse mean curvature flow