The analysis seminar will be held on Friday at 2:30-3:30.
If you are interested in giving a talk contact Gareth Speight (Gareth.Speight<at>uc.edu).
To find out more about our department click here.
Andrew Lorent
Friday September 10 at 2:30-3:30 in French Hall 4206
Rigidity of a non-elliptic differential inclusion related to the Aviles-Giga conjecture
After surveying the topic of regularity of differential inclusions we describe as recent result with Xavier Lamy, Guanying Peng where we prove sharp regularity/rigidity for a differential inclusion into a set K\subset \mathbb{R}^{2\times 2} that arises in connection with the Aviles-Giga functional. The set K is not elliptic, and in that sense our main result goes beyond Šverák's regularity theorem on elliptic differential inclusions and it to our knowledge the first regularity result for non-elliptical differential inclusions. The proof uses the concept of "entropies" that (originates from conservation laws) which allows us to reformulate the issues in terms Fourier multipliers, ultimately the proof comes down to the fact that the Hilbert transform is not bounded from C^0(\mathbb{S}^1) to L^{\infty}(\mathbb{S}^1). We will briefly sketch how this elementary question about a differential inclusion can be resolved using these tools.
Ryan Gibara
Friday September 17 at 2:30-3:30 in French Hall 4211
A (non-exhaustive) list of things I do not know about BMO
In this talk, I am going to explain a few of the many, many things that I do not know about the space BMO of locally integrable functions having bounded mean oscillation on all cubes. Specifically, I will be looking at boundedness and continuity of certain operators on BMO, with an emphasis on sharp constants. A running theme will be how the underlying geometry of cubes, and how they relate to other shapes, plays a role in various inequalities.
Almaz Butaev
Friday October 1 at 2:30-3:30 in French Hall 4206
On BMO extension domains: equivalent definitions and two theorems of Gehring and Osgood
Abstract available here.
Liza Ihnatsyeva (Kansas State University)
Friday October 8 at 2:30-3:30 in West Charlton 240
Hardy-Sobolev inequalities and weighted capacities
Let U be an open set in a metric measure space X. We consider weighted Hardy-Sobolev inequalities in U for the Newtonian (Sobolev) functions. In particular, we study the relation between the validity of the Hardy-Sobolev inequality in U and a quasiadditivity property of a weighted relative capacity. Recall that capacities are (typically) subadditive set functions, which very seldom enjoy full additivity. Quasiadditivity is a weak converse of the subadditivity, involving a multiplicative constant and applicable only to certain types of sets, often given in terms of Whitney-type covers.
This is a joint work with Juha Lehrbäck and Antti V. Vähäkangas.
Vincent Martinez
Friday October 15 at 2:30-3:30 in 60W Charlton 240
On unique ergodicity, regularity, and mixing for the weakly-damped stochastic KdV equation
We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastically forced Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish elementary proofs of both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, a one-force, one-solution principle is established, from which we are able to deduce the existence of a spectral gap with respect to a Wasserstein distance-like function. Our treatment is paradigmatic and should be applicable to many other weakly dissipative systems that are stochastically forced in this way.
Leonid Slavin
Friday October 29 at 2:30-3:30 in French Hall 4206
Sharp dyadic estimates via non-infinitesimal Bellman functions
I will discuss a family of dyadic inequalities involving A_2 weights for which the usual method of obtaining Bellman functions -- solving an appropriate PDE -- fails to yield an answer. This happens because all such PDEs rely on the assumption that the optimizing sequences possess infinitesimal splits (meaning the associated dyadic martingale has arbitrarily small differences) whereas in this setting the splits are, in fact, non-infinitesimal. We construct special Bellman majorants which happen to be the actual Bellman functions for a selection of points in the domain, and thus yield the best possible constants in the inequalities. This is joint work with Brandon Sweeting.
No seminar Nov 5
Mario Bonk (Taft Talk)
Friday November 12 at time TBA
No Seminar Nov 19
Thanksgiving Nov 26