Analysis and Partial Differential Equations

Abstract

The classical Weyl pseudodifferential calculus is a particular choice of “quantisation”– a way to take functions of the position and momentum operators on ${ℝ}^n$. This pseudodifferential calculus allows study of complicated operators to be (mostly) encapsulated by studying their symbols.

Ornstein-Uhlenbeck (OU) operators are analogs of the Laplacian adapted to spaces with Gaussian measure, and arise in many areas including stochastic analysis, quantum field theory and harmonic analysis. They are particularly nasty if tackled analytically and directly, with very rigid structures (for example, the standard OU operator has only $H^{\infty }$ calculus on $L^p$, the proof of which takes over 100 pages in full detail!). From one of these origins, the OU operator arises naturally as a ”function” of position- and momentum-like operators, which suggests that the ideas of Weyl calculi may be applicable.

After explaining these and other relevant concepts, I will explain my current work in adapting the Weyl pseudodifferential calculus to the OU setting. This is of a very different flavour to the standard Weyl pseudodifferential calculus. At present, it seems that using the Weyl calculus splits the problem of studying OU operators into an algebraic part and an analytic part, the analytic part being almost trivial when compared to the analysis used for studying OU directly.

Back to Program

Abstract

Seismometer measurements on the Ross Ice Shelf have confirmed the presence of ocean wave induced vibrations by various components of ocean waves—from the longer tsunami-infragravity waves to the shorter ocean-swell waves. Mathematical models have been developed assuming that the incident wavelengths are much greater than the ice-shelf thickness and the ocean depth. These models make use of the Euler-Bernoulli beam theory for the ice coupled with the linear shallow water equations for the fluid motion, assuming uniform sea-bed. However, the shallow water assumptions are generally valid in the infragravity regime but not valid in the open ocean for ocean-swell waves as the wavelengths are comparable to the ocean depth.

In this talk, we discuss the mathematical model where the thin-beam assumption and the shallow-water conditions are relaxed by assuming full linear-elasticity equations for the ice and the potential flow model for the fluid, respectively. The coupled problem is then solved using the finite element method and the responses of the ice shelf in the frequency-domain and the time-domain are investigated. We will focus in particular on the analytic properties of the solution and show that the solution we calculate can be extended analytically to the entire complex plane. We show further that the solution is dominated by the singularities and zeros in the upper and lower half complex plane.

Back to Program

Abstract

In 1992, Sternberg, Williams and Ziemer showed that on a bounded Lipschitz domain $\Omega$ for every continuous boundary data $g$ the problem

has a unique solution. Here $|Du{|}_{\left(\Omega \right)}$ denotes the total variation of the vector-valued measure $Du$ evaluated on $\Omega$. Characterization of the notion of solutions of the above non-smooth variational problem, were firstly given by Andreu, Ballester, Caselles and Mazón [JFA01]. If the boundary data $\phi$ merely belongs to $L^{\infty }$ then the uniqueness fails to hold (see Mazón et al [Indiana14]). In particular, any kind of continuous dependence for non-smooth boundary data is not known caused by the lack of compactness in BV spaces (cf Adimurthi & Tintarev or Górny).

In this talk, we show how the map assigning Dirichlet data to the co-normal derivative of solutions to the Total Variational Flow ($1$-Laplacian) generates a strongly continuous semigroup on $L^1\left(\partial \Omega \right)$ (and in $L^q$ for all $q\ge 1$. In particular, we show well-posedness and several other important properties of the elliptic-parabolic problem

Here, $\nu$ denotes the outward pointing unit normal vector on $\partial \Omega$. To obtain well-posedness for this elliptic-parabolic problem, the continuous dependence of solutions of the elliptic problem with respect to boundary data is usually used. We outline in this talk how we could circumvent this with functional analytical tools.

The results presented in this talk are obtain in joint work with José Mazón (Universitat de València, Valencia, Spain).

Back to Program

Abstract

It is well-known that the commutator of Riesz transform (Hilbert transform in dimension $1$) and a symbol $b$ is bounded on $L^2\left({ℝ}^n\right)$ if and only if $b$ is in the $BMO$ space $BMO\left({ℝ}^n\right)$ (Coifman–Rochberg–Weiss).

Inspired by this result, it is natural to ask whether it holds for commutator of Riesz transform on Heisenberg groups ${ℍ}^n$ . Note that in the setting of several complex variables, the Heisenberg group ${ℍ}^n$ is the boundary of the Siegel upper half space, whose roles are holomorphically equivalent to the unit sphere and the unit ball in ${ℂ}^n$ respectively, and hence the role of Riesz transform on ${ℍ}^n$ is similar to that of Hilbert transform on the real line $ℝ$.

We answer this question in the setting of stratified Lie groups $G$, which is more general than the Heisenberg group ${ℍ}^n$. We first obtain a suitable version of lower bound for the kernel of the Riesz transform on $G$, and then establish a characterisation for the boundedness of the Riesz commutator, that is, the commutator of Riesz transform and a symbol $b$ is bounded on $L^2\left(G\right)$ if and only if $b$ is in the $BMO$ space $BMO\left(G\right)$ studied by Folland and Stein. In the mean time we also establish characterisations for the endpoint boundedness of Riesz commutators, including the weak type $\left(1,1\right)$, $H^1\left(G\right)\to L^1\left(G\right)$, and $L^{\infty }\left(G\right)$ to $BMO\left(G\right)$, where $H^1\left(G\right)$ is the Hardy space studied by Folland and Stein.

As applications of the Riesz commutators, we introduce the curl operator on $G$ and then establish the div-curl lemma with respect to $H^1\left(G\right)$.

The results we provide here are based on joint work with Xuan Thinh Duong, Hong-Quan Li and Brett D. Wick.

Back to Program

Abstract

In this talk, we introduce a class of noncompact $L_p$-Minkowski problems, and prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampère equations subject to certain boundary conditions depending on the value of $p$. The special case of $p=1$ was previously studied by Pogorelov and Chou-Wang. Here, we give some sufficient conditions for the solvability for general $p$’s.

This is joint work with Yong Huang at Hunan University.

Back to Program

Abstract

Lax-Philips scattering theory is a method to solve for scattering as an expansion over the singularities of the analytic extension of the scattering problem to complex frequencies. I will show how a complete theory can be developed in the case of simple scattering problems. I will illustrate how this theory can be used to find a numerical solution, and I will illustrate the method by applying it to the vibration of ice shelves.

Back to Program

Abstract

Let $\left(X,d,\mu \right)$ be a metric-measure space of homogeneous type, which means that the doubling condition

holds with $C$ independent of $x\in X$ and $r>0$. Here $B\left(x,r\right)$ is a ball in $X$. We shall consider semigroups $T_t$ on $L^p\left(X\right)$, that have integral kernels $T_{t}f\left(x\right)={\int \; <!--nolimits-->}_X{T}_t\left(x,y\right)f\left(y\right)d\mu \left(y\right)$ satisfying Gaussian bounds

It the talk we shall discuss Hölder estimates for $T_t\left(x,y\right)$ which follow from (1). Using this we characterize the Hardy space

by means of atomic decompositions. This means that a function $f$ is in $H^1\left(X,T_t\right)$ if and only if it can be written as $f\left(x\right)={\sum }_k{\lambda }_k{a}_k\left(x\right)$, where ${\sum }_k\left|{\lambda }_k\right|<\infty$ and $a_k\left(x\right)$ are $\left(X,\omega \mu \right)$-atoms. Here $a$ is an $\left(X,\omega \mu \right)$-atom if

where $\omega :X\to \left[C^{-1},C\right]$ is a bounded harmonic function for $T_t$.

This is joint work with Jacek Dziubański.

Back to Program

Abstract

We show that initial closed curves suitably close to a circle flow under the length constrained curve diffusion to round circles in infinite time. We also provide an estimate on the total length of time for which such curves are not strictly convex.

Back to Program

Abstract

We investigate a class of zero-sum linear-quadratic stochastic differential games on a finite time horizon governed by multiscale state equations. The multiscale nature of the problem can be leveraged to show that, for small enough $\epsilon$, the existence of solution to the associated generalised Riccati equation is guaranteed by the existence of a solution to a decoupled pair of differential and algebraic Riccati equations with a reduced order of dimensionality. As a consequence, we are able to construct a couple of asymptotic estimates to the closed-loop value of the game in the sense of an approximate closed-loop strategy, and a limiting value.

Back to Program