During 2017-2018 Spring Semester the seminar will usually be held on Tuesday at 4-5pm in 60 West Charlton Room 240. Everyone is welcome!

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.

2017-2018

Tuesday April 17th, 4-5pm, 60WCharl240: Chandan Biswas (University of Cincinnati)

Extremizers for averaging operators

For a bounded operator a point in the domain is called an extremizer if the operator norm is attained at this point. We study the existence and behavior of extremizers for a model generalized radon transform (also known as averaging operators) and its relation to a restricted X-ray transform.

Friday April 13th, 12:20-1:20pm, Room 60WCHARL 250: Luca Capogna (Worcester Polytechnic Institute)

A Liouville type theorem in sub-Riemannian geometry, and applications to several complex variables

Abstract: The Riemann mapping theorem tells us that any simply connected planar domain is conformally equivalent to the disk. This provides a classification of simply connected domains via conformal maps. This classification fails in higher dimensional complex spaces, as already Poincare' had proved that bi-discs are not bi-holomorphic to the ball. Since then, mathematicians have been looking for criteria that would allow to tell whether two domains are bi-holomorphic equivalent. In the early 70's, after a celebrated result by Moser and Chern, the question was reduced to showing that any bi-holomorphism between smooth, strictly pseudo-convex domains extends smoothly to the boundary. This was established by Fefferman, in a 1974 landmark paper.

Since then, Fefferman's result has been extended and simplified in a number of ways. About 10 years, ago Michael Cowling conjectured that one could prove the smoothness of the extension by using minimal regularity hypothesis, through an argument resting on ideas from the study of quasiconformal maps. In its simplest form, the proposed proof is articulated in two steps: (1) prove that any bi-holomorphism between smooth, strictly pseudoconvex domains extends to a homeomorphisms between the boundaries that is 1-quasiconformal with respect to the sub-Riemannian metric associated to the Levi form; (2) prove that any $1-$quasiconformal homeomorphism between such boundaries is a smooth diffeomorphism.

In this talk I will discuss recent work with Le Donne, where we prove the first step of this program, as well as joint work with Citti, Le Donne and Ottazzi, where we settle the second step, thus concluding the proof of Cowling's conjecture. The proofs draw from several fields of mathematics, including nonlinear partial differential equations, and analysis in metric spaces.

Tuesday April 10th, 4-5pm, 60WCharl240: Robert Wolf (University of Cincinnati)

Compactness of Iso-resonant Potentials

Abstract: Bruning considered sets of isospectral Schrodinger operators with smooth real potential on a compact manifold of dimension 3. He showed the set of potentials associated to an isospectral set is compact in the C infinity topology by relating the spectrum to the trace of the heat semi-group. Similarly, we can consider the resonances of Schrodinger operators with real valued potentials on R3 whose support lies inside a ball of fixed radius that generate the same resonances as some fixed Schrodinger operator, an "isoresonant" set of potentials. Using the Poisson formula to relate the resonances to the trace of the wave group, we can show that this "isoresonant" set of potentials is also compact the C infinity topology.

Tuesday March 27th, 4-5pm, 60WCharl240: Andrew Lorent (University of Cincinnati)

An absolute three well Liouville Theorem

Abstract: It is a consequence of Liouville's theorem that differential inclusions into the set of rotations are affine. Somewhat surprisingly an optimal quantitative version of this corollary is true and has been proved 15 years ago by Friesecke, James, Muller. This inspired a great deal of research including work with R.L. Jerrard on an optimal quantitative version for differential inclusions into two copies of the set of rotations, a "Two well Liouville theorem". Easy counterexamples show there can not exist a four well Liouville theorem but a three well theorem is unknown even in absolute case. The theorem I will discuss is the first absolute three well Liouvile theorem. This whole area is somewhat motivated by the Ball-James, Chipot-Kinderleher theory of phases transitions in crystalline solids via non convex Calculus of Variations, we will explain the background and previous results at length and briefly touch on the three well theorem and its methods of proof at the end of the talk.

Tuesday March 20th, 4-5pm, 60WCharl240: Xiaojun Wang (University of Cincinnati)

Stability of the solitary wave solutions to a coupled BBM system

Abstract: We will review the concept of stability for solitary wave and discuss a result we got for a coupled BBM system. The result takes advantage of the accurate point spectrum information of the associated Schrodinger operator.

Tuesday March 6th, 4-5pm, 60WCharl240: Leonid Slavin (University of Cincinnati)

Dimension-free estimates for semigroup BMO and related classes

Tuesday February 27th, 4-5pm, 60WCharl240: Bingyu Zhang (University of Cincinnati)

Kato Smoothing and Sharp Kato Smoothing Properties of Dispersive Wave Equations on a Periodic Domain (Part 2)

Tuesday February 20th: no seminar (colloquium at 4-5pm in West Charlton 135)

Tuesday February 13th, 4-5pm, 60WCharl240: Bingyu Zhang (University of Cincinnati)

Kato Smoothing and Sharp Kato Smoothing Properties of Dispersive Wave Equations on a Periodic Domain

Tuesday February 6th: no seminar

Tuesday January 30th, 4-5pm, 60WCharl240: Michael Goldberg (University of Cincinnati)

Fourier restriction estimates for derivatives

Abstract: As a general rule, the Fourier transform of a function in L^p(R^n) belongs to the dual space L^{p'}(R^n) -- if p is between 1 and 2 -- but it need not be differentiable anywhere in any way. We look for conditions under which a derivative exists for points along a curved surface with codimension 1.

It is worth keeping in mind that functions in L^{p'}(R^n) are only defined almost everywhere. Establishing the existence of a restriction to a lower-dimensional surface is already a nontrivial task exemplified by the Stein-Tomas restriction theorem. We use similar tools from functional analysis to construct an operator representing the derivative of the Fourier transform, then show that it is bounded on a natural (yet surprising) subset of L^p.

This is joint work with Dmitriy Stolyarov.

Tuesday January 23rd, 4-5pm, 60WCharl240: Taige Wang (University of Cincinnati)

Analysis and simulations on a shear banded fluid

Abstract: The PEC (partially extending strand convection) model of Larson is able to describe thixotropic yield stress behavior in the limit where the relaxation time is large relative to the retardation time. In this talk, we discuss the development of shear bands in planar Poiseuille flow which is started up from rest with an imposed pressure gradient. We analyze the asymptotic limit of large relaxation time; the small parameter ε measures the ratio of retardation time to relaxation time. We determine the position and width of shear bands as a function of time. We identify an initial phase of ”fast yielding” during which the width of the transition between high and low shear rate regions behaves like t^−3. This continues until t (measured on the scale of the retardation time) is on the order of ε^−1/3. Then there is a phase of `delayed yielding' during which the width of the transition is of order ε. Eventually, the width sharpens as 1/(ε^2t^3).

Thursday November 30th, 4-5pm, seminar room: Xiaodan Zhou (Worcester Polytechnic Institute)

Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group

Abstract: In the Euclidean space, Lipschitz continuity and convexity preserving are two very important properties, closely related to the maximum or comparison principle, which hold for a large class of linear and nonlinear parabolic equations. In this talk, we intend to extend these preserving properties to nonlinear equations in the Heisenberg group $\mathbb{H}$. However, these generalization are by no means immediate. In fact, our counterexamples show that preserving of Lipschitz continuity and horizontal convexity may fail even for very simple linear equations. On the other hand, there are many examples on Lipschitz and convexity preserving in the Heisenberg group. We prove Lipschitz and horizontal convexity preserving properties under appropriate assumptions. This is joint work with Qing Liu and Juan Manfredi.

Thursday November 16th, 4-5pm, seminar room: Andrew Lorent (University of Cincinnati)

Null Lagrangian measures, Compensated compactness and Conservation laws (Part 2)

Abstract: Compensated compactness is one of the main methods for solving non-linear PDE. In particular hyperbolic conservation laws, at their heart - these proofs come down to showing a particular class of measures (call Null Lagrangian measures) supported on submanifolds in matrix space are actually Dirac measures. A general necessary and sufficient characterization of the conditions under which this is true is unknown even for planes in Matrix space, we present results in this direction, an application to a question of Kircheim, Muller, Sverak and apply our results reformation of a classic result in conservation laws. This is joint work with Guanying Peng.

Thursday November 9th, 4-5pm, department colloquium in 120 W Charlton

Thursday November 2nd, 4-5pm, seminar room: Andrew Lorent (University of Cincinnati)

Null Lagrangian measures, Compensated compactness and Conservation laws

Abstract: Compensated compactness is one of the main methods for solving non-linear PDE. In particular hyperbolic conservation laws, at their heart - these proofs come down to showing a particular class of measures (call Null Lagrangian measures) supported on submanifolds in matrix space are actually Dirac measures. A general necessary and sufficient characterization of the conditions under which this is true is unknown even for planes in Matrix space, we present results in this direction, an application to a question of Kircheim, Muller, Sverak and apply our results reformation of a classic result in conservation laws. This is joint work with Guanying Peng.

Thursday October 26th, 4-5pm, seminar room: Dimitrios Ntalampekos (UCLA)

Harmonic functions on Sierpinski carpets

Abstract: I will discuss a notion of Sobolev spaces and harmonic functions on Sierpinski carpets, which differs from the classical approach of potential theory in metric measure spaces. The goal is to define a notion that takes into account also the ambient space, where the carpet lives. As an application of carpet-harmonic functions we obtain a quasisymmetric uniformization result for Sierpinski carpets.

Thursday October 19th, 4-5pm, department colloquium in 120 WCharlton: Shuming Sun (Virginia Tech)

Exact theory on existence of solitary or multi-solitary water waves

Abstract: The talk will discuss recent development on the existence of two- and three-dimensional solitary or multi-solitary surface waves on the water of finite depth with or without surface tension using the exact governing equations (called Euler equations). It will be shown that when the nondimensional wave-speed and surface tension are in various regions, the Euler equations possess several different kinds of two- or three-dimensional solitary or multi-solitary wave solutions. Moreover, some stability results for these waves will be addressed, such as transverse instability, spectral stability, asymptotic linear stability or conditional stability. The talk is accessible to non-experts or graduate students.

Thursday October 12th, 4-5pm, seminar room: Xin Yang (University of Cincinnati)

Estimate of the lifespan for the heat equation with local nonlinear Neumann boundary conditions

Abstract: This talk is about the blow-up problem for the heat equation $u_t=\Delta u$ in a $C^{2}$ bounded open subset $\Omega$ of $\m{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a local nonlinear Neumann boundary condition: the normal derivative $\frac{\p u}{\p n}=u^{q}$ on partial boundary $\Gamma_1\subseteq\p\O$ for $q>1$ and $\frac{\p u}{\p n}=0$ on the rest of the boundary. The motivation of this problem is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. It has been known that the classical solution blows up in finite time. The focus of this talk is on the estimate of the lifespan $T^{*}$ of the solution in terms of the nonlinearity $q$, the maximum $M_{0}$ of the initial data and the surface area $|\Gamma_{1}|$ of $\Gamma_{1}$. In particular, the asymptotic behaviour of $T^{*}$ as $q\rightarrow 1 $, $M_{0}\rightarrow 0$ or $|\Gamma_{1}|\rightarrow 0$ will be discussed. This is joint work with Zhengfang Zhou.

Thursday October 5th, 4-5pm, department colloquium in 119 WCharlton: Fabrice Baudoin (University of Connecticut)

Laplacian comparison theorems in sub-Riemannian geometry

Abstract: We will discuss Laplacian comparison theorems in sub-Riemannian geometry. The main idea in this area is to approximate the sub-Riemannian space by Riemannian structures satisfying a uniform measure contraction property.

Thursday September 28th, 4-5pm, seminar room: Gareth Speight (University of Cincinnati)

Maximal directional derivatives and universal differentiability sets in Carnot groups

Abstract: Rademacher's theorem asserts that Lipschitz functions from R^n to R^m are differentiable almost everywhere. Investigating converses to Rademacher's theorem leads to the following result: if n>1 then there exists a Lebesgue null set N in R^n containing a point of differentiability for every Lipschitz mapping f:R^n->R. Such sets are called universal differentiability sets and their construction relies on the fact that existence of an (almost) maximal directional derivative implies differentiability. We discuss recent work extending these results to Carnot groups of different step. Based on joint work with Andrea Pinamonti and Enrico Le Donne.

Thursday September 21st, 4-5pm, seminar room: Vito Buffa (University of Ferrara)

BV functions in Metric Measure Spaces: new insights on integration by parts formula, and traces

Abstract: We make use of the differential structure developed by N. Gigli in a recent paper in order to provide a definition of BV functions via the divergence of suitable vector fields, showing that an integration by parts formula holds also for open domains. A Gauss-Green Formula is established, characterizing the normal trace of "divergence-measure" vector fields. We then discuss traces of BV functions re-adapting the theory of "rough traces" (after V. Maz'ya) to the metric setting, in comparison with the "Lebesgue-point" approach. Based on joint work with Michele Miranda.

Thursday September 14th, 4-5pm, seminar room: Tom Hill (University of Cincinnati)

A model of thermal explosion in porous media

Abstract: In this paper we consider a model of thermal explosion in porous media. The model consists of two reaction-diffusion equations in a bounded domain with Dirichlet boundary conditions and describes the initial stage of evolution of pressure and temperature fields. Under certain conditions, the classical solution of these equations exists only on finite time interval after which it forms a singularity and becomes unbounded (blows up). This behavior raises a natural question whether this solution can be extended, in a weak sense, after blow up time. We prove that the answer to this question is no, that is, the solution becomes unbounded in entire domain immediately after the singularity is formed. From a physical perspective our results imply that autoignition in porous materials occurs simultaneously in entire domain.

Thursday September 7th, 4-5pm, seminar room: Michele Miranda (University of Ferrara)

Surface measures in Wiener Spaces

Abstract: In the present talk we shall give an overview on recent developments in the theory of surface measure; we in particular shall see the measures introduced by Airault-Malliavin, the Hausdorff measures introduced by Feyel-De La Pradelle and the perimeter measures. We are in particular interested in the connections among these measures and the validity of integration by parts formulae in Wiener spaces.