## 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

$$\left\{\begin{array}{cc}{B}_{1-2s}u\ni 0\phantom{\rule{1em}{0ex}}\hfill & \text{in}{H}_{+}:=H\times \left(0,+\infty \right)\text{,}\hfill \\ u=\phi \phantom{\rule{1em}{0ex}}\hfill & \text{on}\partial {H}_{+}=H\text{,}\hfill \end{array}\right.$$ | (${D}_{\phi}^{s}$) |

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
$\phi \in {\overline{D\left(A\right)}}^{{\text{}}_{H}}$. This
enables us to investigate the Dirichlet-to-Neumann (D-t-N) operator

$$\phi \mapsto {\Lambda}_{s}\phi :=-\underset{t\to 0+}{lim}{t}^{1-2s}{u}_{t}\left(t\right)$$

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
$s=1\u22152$ 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.