PDE Seminar Abstracts

Fractional powers of monotone operators in Hilbert spaces

Daniel Hauer
University of Sydney
Mon 27th May 2019, 2-3pm, Carslaw Room 829 (AGR)


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(A) of A, then for every 0 < s < 1, the Dirichlet problem

B1-2su 0  in H+ := H × (0, +), u = φ on H+ = H, (Dφs)

associated with the Bessel-type operator B1-2su := -1-2s t ut - utt + Au is well-posed for every boundary value φ D(A)¯H. This enables us to investigate the Dirichlet-to-Neumann (D-t-N) operator

φΛsφ := - lim t0+t1-2su t(t)

on H (where u solves (Dφs)) associated with B1-2s and to define the (Ls2) fractional power As of A via the extension problem (Dφs). We investigate the semigroup {e-Ast} t0 generated by - As on H; prove comparison principles, contractivity properties of {e-Ast} t0 in Orlicz spaces Lψ, and show that As admits a sub-differential structure provided A has it as well.

The results extend earlier ones obtained in the case s = 12 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 B1-2s, 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.