**SMS scnews item created by Daniel Daners at Fri 24 May 2019 1129**

Type: Seminar

Modified: Fri 24 May 2019 1149; Fri 24 May 2019 1152

Distribution: World

Expiry: 27 May 2019

**Calendar1: 27 May 2019 1400-1500**

**CalLoc1: AGR Carslaw 829**

CalTitle1: Hauer: Fractional powers of monotone operators in Hilbert spaces

Auth: daners@dora.maths.usyd.edu.au

### PDE Seminar

# Fractional powers of monotone operators in Hilbert spaces

### Hauer

Daniel Hauer

University of Sydney

Mon 27th May 2019, 2-3pm, Carslaw Room 829 (AGR)

## 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.

For Seminar announcements you can now subscribe to the Seminar RSS feed.
Check also the PDE Seminar page.

Enquiries to Daniel Hauer or Daniel Daners.

**Actions:**

Calendar
(ICS file) download, for import into your favourite calendar application

UNCLUTTER
for printing

AUTHENTICATE to mark the scnews item as read