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 is a maximal monotone operator on a Hilbert space with in the range of , then for every , the Dirichlet problem
associated with the Bessel-type operator is well-posed for every boundary value . This enables us to investigate the Dirichlet-to-Neumann (D-t-N) operator
on (where solves ()) associated with and to define the () fractional power of via the extension problem (). We investigate the semigroup generated by on ; prove comparison principles, contractivity properties of in Orlicz spaces , and show that admits a sub-differential structure provided has it as well.
The results extend earlier ones obtained in the case 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 , 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.
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