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 \(\textrm{Rg}(A)\) of \(A\), then for every \(0 < s < 1\), the Dirichlet problem associated with the Bessel-type equation \[ A_{1-2s}u:= -\frac{1-2s}{t}u_{t}-u_{tt}+Au\ni 0 \] is well-posed for boundary values \(\varphi\in \overline{D(A)}^{\mbox{}_{H}}\). This allows us to define the Dirichlet-to-Neumann (DtN) map \(\Lambda_{s}\) associated with \(A_{1-2s}\) as \[ \varphi\mapsto \Lambda_{s}\varphi:=-\lim_{t\to 0+}t^{1-2s}u_{t}(t)\qquad\text{in H.} \] The existence of the DtN map \(\Lambda_{s}\) associated with \(A_{1-2s}\) is the first step in defining fractional powers \(A^{\alpha}\) of monotone (possibly, nonlinear and multivalued) operators \(A\) on \(H\). We prove that \(\Lambda_{s}\) is monotone on \(H\), and if \(\overline{\Lambda}_{s}\) is the closure of \(\Lambda_{s}\) in \(H\times H_{w}\) then we provide conditions implying that \(-\overline{\Lambda}_{s}\) generates a strongly continuous semigroup on \(\overline{D(A)}^{\mbox{}_{H}}\). In addition, we show that if \(A\) is completely accretive, on \(L^{2}(\Sigma,\mu)\) for a \(\sigma\)-finite measure space \((\Sigma,\mu)\), then \(\Lambda_{s}\) inherits this property from \(A\).

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