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September  2015, 14(5): 2043-2067. doi: 10.3934/cpaa.2015.14.2043

## A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains

 1 University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received  January 2015 Revised  April 2015 Published  June 2015

Let $\Omega\subset R^N$ be a bounded open set with Lipschitz continuous boundary $\partial \Omega$. We define a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on $L^2(\partial \Omega)$ which can also be ultracontractive. We also use the fractional Dirichlet-to-Neumann operator to compare the eigenvalues of a realization in $L^2(\Omega)$ of the fractional Laplace operator with Dirichlet boundary condition and the regional fractional Laplacian with the fractional Neumann boundary conditions.
Citation: Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043
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