Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 223–269 | Cite as

On a fractional (sp)-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.

Keywords

Fractional p-Laplacian Fractional p-Dirichlet-to-Neumann operator Nonlinear submarkovian semigroup Ultracontractivity of (nonlinear) semigroups Nonlocal quasi-linear elliptic problems Regularity of weak solutions 

Mathematics Subject Classification

35R11 35B65 35K65 47H20 35B40 

Notes

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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Copyright information

© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics (Rio Piedras Campus), Faculty of Natural SciencesUniversity of Puerto RicoSan JuanUSA

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