Fractional powers of monotone operators in Hilbert spaces

Abstract

In this talk I want to present a functional analytical framework for defining fractional powers of maximal monotone (possibly, multi-valued and nonlinear) operators in Hilbert spaces. We begin by showing that if $A$ is a maximal monotone operator on a Hilbert space $H$ with $0$ in the range $Rg\left(A\right)$ of $A$, then for every $0<s<1$, the Dirichlet problem

associated with the Bessel-type operator $B_{1-2s}u:=-\frac{1-2s}{t}{u}_t-u_{tt}+Au$ is well-posed for every boundary value $\phi \in {\overline{D\left(A\right)}}^{{}_H}$. This enables us to investigate the Dirichlet-to-Neumann (D-t-N) operator

on $H$ (where $u$ solves ($D_{\phi }^s$)) associated with $B_{1-2s}$ and to define the ($L_s^2$) fractional power $A^s$ of $A$ via the extension problem ($D_{\phi }^s$). We investigate the semigroup ${\left\{e^{-A^{s}t}\right\}}_{t\ge 0}$ generated by $-A^s$ on $H$; prove comparison principles, contractivity properties of ${\left\{e^{-A^{s}t}\right\}}_{t\ge 0}$ in Orlicz spaces $L^{\psi }$, and show that $A^s$ admits a sub-differential structure provided $A$ has it as well.

The results extend earlier ones obtained in the case $s=1∕2$ by Brezis [Israel J. Math. 72], Barbu [J. Fac. Sci. Univ. Tokyo Sect. IA Math.72.].

As a by-product of the theory developed in the presented work, we also obtain well-posedness of the Robin problem associated with $B_{1-2s}$, which might be of independent interest.

The results are joint work with the two former undergraduate student Yuhan He and Dehui Liu during summer 2018 at the University of Sydney.