司靈得 (Daniel Spector)

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.

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.

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.

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.

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.

In 1980, C. Kenig proved that for every Lipschitz domain Ω in the plane there exists 1 ≤ p₀ < 2 so that the Dirichlet problem has a solution for every f ∈ L^p(∂Ω) and every p ∈ (p₀, ∞). Moreover, if p₀ > 1, the result is false for p ≤ p₀. The goal of this talk is to analyze what happens at the endpoint p₀; that is, we want to look for spaces X ⊂ L^p₀ so that the Dirichlet problem has a solution for every f ∈ X. These spaces X will be either a Lorentz space L^p₀,1(∂Ω) or some Orlicz space of logarithmic type. Similar results will be presented for the Neumann problem. This is a joint work with Virginia Naibo and Carmen Ortiz-Caraballo.

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.

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.