Abstract: In this talk I
will introduce a type of Sobolev multiplier which appears
naturally in many super critical nonlinear PDEs. We will
briefly study the preduals of the Sobolev multplier spaces
and the boundedness of Hardy-Littlewood maximal operators on
such spaces. Furthermore, the boundedness of maximal
operators on the spaces of Choquet integrals associated with
capacities will also be addressed. The main tools in
tackling these problems rely on classical harmonic analysis
and nonlinear potential theory.
Tuesday 21th March 2023,
3:00–4:00 pm TST (GMT+8), online on Zoom
Abstract:
The porous medium equation is a degenerate
parabolic type quasilinear equation that models, for
example, the flow of a gas through a porous medium. In this
talk I will present recent results on uniqueness in the
inverse boundary value problem for this equation. These are
the first such results to be obtained for a degenerate
parabolic equation. The talk is based on work with T. Ghosh
& G. Nakamura and T. Ghosh & G. Uhlmann.
Tuesday 28th March 2023,
3:00–4:00 pm TST (GMT+8), online on Zoom
Abstract: Optimal
embeddings among fractional Orlicz-Sobolev spaces with
different smoothness are characterized. The equivalence of
their Gagliardo-Slobodeckij norms to norms defined via
Littlewood-Paley decompostions, via oscillations, or via
Besov type difference quotients is also established. These
equivalences, of independent interest, are a key tool in the
proof of the relevant embeddings. This is joint work with
Andrea Cianchi.
Tuesday 28th March 2023,
3:00–4:00 pm TST (GMT+8), online on Zoom
Abstract: Optimal
embeddings among fractional Orlicz-Sobolev spaces with
different smoothness are characterized. The equivalence of
their Gagliardo-Slobodeckij norms to norms defined via
Littlewood-Paley decompostions, via oscillations, or via
Besov type difference quotients is also established. These
equivalences, of independent interest, are a key tool in the
proof of the relevant embeddings. This is joint work with
Andrea Cianchi.
Tuesday 11th April 2023,
3:00–4:00 pm TST (GMT+8), online on Zoom
Abstract:
In this talk we discuss how to extend the classical
Bernstein technique to the setting of integro-differential
operators. As a consequence of this, we are able to provide
first and one-sided second derivative estimates for
solutions to fractional equations. Our method is robust
enough to be applied to some Pucci-type extremal equations
and to obstacle problems for fractional operators.
Abstract:
In this talk, we discuss some bilinear
fractional integral operators introduced by Kenig and Stein.
Also, the Stein-Weiss inequality and its bilinear analogues
will be addressed in Euclidean space and beyond. This is a
joint work with Rajesh K. Singh.
Abstract: We study
different problems with energy gaps: local and nonlocal
double potential, variable exponent and weights models. We
design the general procedure to construct new examples of
energy gaps and present the numerical scheme that converges
to the global minimiser of the problem. The talk is based on
several joint projects with Lars Diening, Michail Surnachev,
Johanness Srorn and Christoph Ortner.
