Preprint

Fractional powers of monotone operators in Hilbert spaces

Daniel Hauer, Yuan He, Dehui Liu


Abstract

In this article, we show that if A is a maximal monotone operator on H with 0 in the range Rg(A) of A, then for every 0<s<1, the Dirichlet problem associated with the Bessel-type equation A12su:=12stututt+Au0 is well-posed for boundary values φD(A)H. This allows us to define the Dirichlet-to-Neumann (DtN) map Λs associated with A12s as φΛsφ:=limt0+t12sut(t)in H. The existence of the DtN map Λs associated with A12s is the first step in defining fractional powers Aα of monotone (possibly, nonlinear and multivalued) operators A on H. We prove that Λs is monotone on H, and if Λs is the closure of Λs in H×Hw then we provide conditions implying that Λs generates a strongly continuous semigroup on D(A)H. In addition, we show that if A is completely accretive, on L2(Σ,μ) for a σ-finite measure space (Σ,μ), then Λs inherits this property from A.

Keywords: Monotone operators, Hilbert space, evolution equations, fractional operators.

This paper is available as a pdf (488kB) file. It is also on the arXiv: arxiv.org/abs/1805.00134.

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