Analysis and Partial Differential Equations

Abstract

Logarithmic Sobolov inequalities arose in quantum field theory and were introduced to describe smoothing properties of Markov semigroups. In 1975 L. Gross proved a log-Sob inequality for the Gaussian measure on ${ℝ}^n$, and the question immediately arose as to just which measures satisfied a similar inequality.

The periodic KdV equation $u_t=u_{xxx}+\beta u{u}_x$ arises from a Hamiltonian system with an infinite dimensional phase space $L^2\left(T\right)$. J. Bourgain showed that there is a Gibb’s measure $\nu ={\nu }_r^{\beta }$ on each closed ball $B_r$ of radius $r$ in $L^2\left(T\right)$ such that the Cauchy problem for this PDE is well-posed on the support of $\nu$, and such that $\nu$ is invariant under the KdV flow.

In this talk I shall discuss some joint work with Gordon Blower (Lancaster) in which we prove that these measures satisfy logarithmic Sobolov inequalities. These log-Sob inequalities are then used to prove concentration inequalities for Lipschitz functions on the balls $B_r$.

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Abstract

Consider the Hermite operator $H=-\Delta +|x{|}^2$ on the Euclidean space ${ℝ}^n$. In this talk, we develop a theory of homogeneous and inhomogeneous Besov and Triebel-Lizorkin spaces associated with the Hermite operator. Applications include the boundedness of negative powers and spectral multipliers of the Hermite operators on some appropriate Besov and Triebel-Lizorkin spaces. This is joint work with The Anh Bui which appeared in Journal of Fourier Analysis and Applications recently (doi:10.1007/s00041-014-9378-6).

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Abstract

Random walks have been used to model stochastic processes in many scientific fields. I will introduce invariant random walks on groups, where the transition probabilities are given by a probability measure. The Poisson boundary will also be discussed. It is a space associated with every group random walk that encapsulates the behaviour of the walks at infinity and gives a description of certain harmonic functions on the group in terms of the essentially bounded functions on the boundary. I will then discuss my attempts to describe the boundary for a certain family of upper-triangular matrix groups.

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Abstract

We present a finite element method for nearly incompressible elasticity using a mixed formulation of linear elasticity in the displacement-pressure form. We combine the idea of stabilization of an equal order interpolation for the Stokes equations with the idea of biorthogonality to get rid of the bubble functions used in an earlier publication with a biorthogonal system. We work with a Petrov-Galerkin formulation for the pressure equation, where the trial and test spaces are different and form a $g$-biorthogonal system. This novel approach leads to a displacement-based low order finite element method for nearly incompressible elasticity for simplicial, quadrilateral and hexahedral meshes. Numerical results are provided to demonstrate the efficiency of the approach.

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Abstract

The Witten index of an operator $T$ can be considered as a substitution for the Fredholm index of $T$, whenever the operator $T$ ceases to be Fredholm. The Witten index is closely related to the notion of the spectral shift function. In particular, if $A$ is a self-adjoint operator on a Hilbert space $H$, $B$ is a self-adjoint bounded operator on $H$ and $\theta$ is a parameter function on $ℝ$, then the Witten index of the operator $D_A=\frac{d}{dt}\otimes 1+1\otimes A+M_{\theta }\otimes B$ on the Hilbert space $L^2\left(ℝ\right)\otimes H$ can be computed as the value of the spectral shift function for the pair $\left(A+B;A\right)$ at zero, where $M_{\theta }$ is the operator given by multiplication by the function $\theta$. However, the assumptions on the operators $A$ and $B$ rules out the classical differential operators even in low dimensions. We generalize the earlier results and compute the actual value of the Witten index of the operator $D_A=\frac{d}{dt}\otimes 1-1\otimes i\frac{d}{dx}+M_{\theta }\otimes M_f$, $dom\left(D_A\right)=W^{1,2}\left({ℝ}^2\right)$ on the Hilbert space $L^2\left({ℝ}^2\right)=L^2\left(ℝ\right)\otimes L^2\left(ℝ\right)$.

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Abstract

In quantum chaos, we make use of tools from harmonic and microlocal analysis to examine how the dynamical features of a classical Hamiltonian system are reflected in the behaviour of the PDE that governs its quantisation.

The quantum ergodicity theorem of Shnirelman, Colin de Verdière, and Zelditch is a cornerstone theorem in this field, which states that if a classical Hamiltonian system has ergodic flow with respect to the Liouville measure, then the quantum evolution satisfies an analogous equidistribution property.

Much less well understood is the quantum evolution of mixed systems, whose phase spaces divide into multiple invariant subsets, only some of which are ergodic. In this talk I will present my recent result which establishes a longstanding conjecture of Percival for the simplest such system, the mushroom billiard.

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Abstract

Finite differences have long been used to approximate derivativeso of functions. It may be less well known that on the first order Sobolev space of the circle group or the real line, the derivative of a function is equal to the sum of three first order differences, and that this number three is sharp – that is, the statement fails with two differences in place of three. Such results involve the behaviour of the Fourier transforms of a function near the origin or, in the case of other differential operators, with the zeros of the Fourier transform of a function on a given subset of the integers, or the behaviour of the Fourier transform near a subset of the real line. In this talk, Hilbert spaces arising from finite sums of differences and generalised differences on the circle group and the real line will be discussed, and some associated sharpness results described.

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Abstract

Riesz transforms have long been studied in mathematical analysis. A common condition for $L^p$ boundedness of Riesz transforms, $p>2$, is a preservation condition. Suppose now that the preservation condition does not hold: $e^{-tL}1\ne 1$, and consider the boundedness of a generalised Riesz transform $\nabla L^{-1∕2}$ defined on this space.

In this talk I will discuss methods of proving $L^p$ boundedness, $p>2$, of such a transform on such a space. These methods are highly focused on heat kernel bounds and various analytical heat kernel properties. Applications will be to subsets of ${ℝ}^n$ and a particular reliance on types of Gaffney and Hardy operators will be discussed.

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Abstract

In the late 1950s, two schools obtained a series of results on non-autonomous linear parabolic equations with bounded measurable (in space and time) coefficients. Lions’ school, on the one hand, used form methods. They could handle systems, but had to work with $L_2$ data. Nash and Aronson, on the other hand, used heat kernel methods, and obtained regularity results that allowed them to handle $L_p$ data. Their method, however, could not be applied to systems. Ever since, results for systems and $L_p$ data have been limited to coefficients that are Holder regular in time.

Pascal Auscher, Sylvie Monniaux, and I have developed, over the past four years, a new approach to the problem that lifts these limitations. For $p\ge 2$, we obtain existence and uniqueness results for systems with bounded measurable coefficients in space and time, and Lp data. For a range of values of $p$ below $2$, we obtain existence and uniqueness results for Lp data, and systems with coefficients that are $BV$ in time and bounded measurable in space.

Our approach has its origin in elliptic boundary value problems on rough domains. We consider our evolution equation as a boundary value problem in a rough space-time domain, and extend singular integrals, functional calculus, and Hardy space methods originally designed for elliptic boundary value problems.

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Abstract

We investigate a periodic-parabolic problem with a parameterised weight function. We are particularly interested in the behaviour of the principal eigenvalue and associated eigenfunction as the parameter goes to infinity. The principal eigenvalue has already been studied by Hess, Du and Peng, but rather than approach the problem directly we instead look at the evolution problem and associated evolution operator. This allows for greater generality and gives more information about the associated positive eigenfunction; in particular, if the weight function has regular enough support, the limiting problem is in some sense just the evolution problem on a restricted, non-cylindrical domain. This has applications to periodic-parabolic logistic-type population problems.

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