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On a fractional (s, p)-Dirichlet-to-Neumann operator on bounded Lipschitz domains

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Abstract

Let \(p\in (1,\infty )\) and \(\Omega \subset \mathbb {R}^{N}\) a bounded open set with Lipschitz continuous boundary \(\partial \Omega \). We define a fractional p-Dirichlet-to-Neumann operator associated with the regional fractional p-Laplace operator \((-\Delta )_{p,\Omega }^{s}\), \(0<s<1\), and prove that it generates a strongly continuous semigroup on \(L^{2}(\partial \Omega )\) which is order preserving and non-expansive on \(L^\infty (\partial \Omega )\). We show the convergence as time goes to \(\infty \) of all the trajectories of the semigroup. Some results of existence, regularity and fine a priori estimates of solutions to elliptic and parabolic problems associated with the fractional p-Dirichlet-to-Neumann operator are also obtained.

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Acknowledgements

We thank the referee for her/his careful reading of the first version of the manuscript and for her/his useful comments that have helped to improve the paper. The work of the author is partially supported by the Air Force Office of Scientific Research under the Award No: FA9550-15-1-0027.

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  1. Department of Mathematics (Rio Piedras Campus), Faculty of Natural Sciences, University of Puerto Rico, 70377, San Juan, PR, 00936-8377, USA

    Mahamadi Warma

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Warma, M. On a fractional (s, p)-Dirichlet-to-Neumann operator on bounded Lipschitz domains. J Elliptic Parabol Equ 4, 223–269 (2018). https://doi.org/10.1007/s41808-018-0017-2

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