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

Abstract

Let Ω⊂ ℝN be a bounded open set with Lipschitz continuous boundary ∂Ω. We define a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on L2(∂Ω) which can also be ultracontractive. We also use the fractional Dirichletto-Neumann operator to compare the eigenvalues of a realization in L2(Ω) of the fractional Laplace operator with Dirichlet boundary condition and the regional fractional Laplacian with the fractional Neumann boundary conditions.