South Wales Analysis and Probability Seminar (SWAP)

13 December 2019 at Cardiff University (poster)

 9:30-10:00 Coffee & Registration 10:00-10:45 Frank Rösler (Cardiff) Norm-resolvent convergence in perforated domains 10:45-11:30 Vitaly Moroz (Swansea) Thomas-Fermi type models of external charge screening in graphene 11:30-11:50 Coffee 11:50-12:35 Jonathan Ben-Artzi (Cardiff) Convergence rates in dynamical systems lacking a spectral gap 12:35-14:00 Lunch Colloquium: 14:00-14:50 Xavier Ros-Oton (Zürich) Generic regularity in free boundary problems 14:50-15:20 Coffee 15:20-16:05 Eugene Lytvynov (Swansea) Orthogonal polynomials of Lévy white noise and umbral calculus

Abstracts

Frank Rösler (Cardiff) - Norm-resolvent convergence in perforated domains

For different types of boundary conditions (Dirichlet, Neumann and Robin), we prove norm-resolvent convergence in L2 for the Laplacian in an open domain perforated epsilon-periodically by spherical holes. The limit operator is of the form -Δ+m on the unperforated domain, where m is a positive constant. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Proc. Japan Acad., 61, Ser. A (1985)], who showed strong resolvent convergence. In particular, our result implies convergence of the spectrum of the operator for the perforated domain problem.

Vitaly Moroz (Swansea) - Thomas-Fermi type models of external charge screening in graphene

Graphene is a recently discovered material which consists of exactly one atomic layer of carbon. We discuss density functional theories of Thomas-Fermi and Thomas-Fermi-von Weizsacker type which describe the response of a single layer of graphene to an external electric charge. Mathematically, this amounts to the analysis of two nonlocal variational problems which involve Coulombic terms and a Hardy type potential. We develop the variational framework in which the proposed energy functionals admit minimizers and prove the uniqueness and regularity of the ground states for the associated Euler-Lagrange equations which involve the fractional Laplacian. In addition, we discuss the decay rate (screening) of the ground states and present several open problems. This is a joint work with Jianfeng Lu and Cyrill Muratov.

Jonathan Ben-Artzi (Cardiff) - Convergence rates in dynamical systems lacking a spectral gap

Our world is neither compact nor periodic. It is therefore natural to consider dynamical systems on unbounded domains, where typically there is no spectral gap. I will present a (simple) method for studying the generators of such systems where a spectral gap assumption is replaced with an estimate of the Density of States (DoS) near zero. There are two main applications:

1) Dissipative systems: when the generator is non-negative, an estimate of the DoS leads to a so-called "weak Poincaré inequality" (WPI). This in turn leads (in some cases) to an algebraic decay rate for the $$L^2$$ norm of the solution. For instance, in the case of the Laplacian (generator of the heat equation) the WPI is simply the Nash inequality which leads to the optimal decay rate of $$t^{-d/4}$$.

2) Conservative systems: when the generator is skew-adjoint, an estimate of the DoS leads to a uniform ergodic theorem on an appropriate subspace. Examples include the linear Schrödinger equation and incompressible flows in Euclidean space.

Based on joint works with Amit Einav (Graz) and Baptiste Morisse (Cardiff).

Xavier Ros-Oton (Zürich) - Generic regularity in free boundary problems

Free boundary problems are those described by PDE's that exhibit a priori unknown (free) interfaces or boundaries. They appear in Geometry, Physics, Probability, Biology, or Finance, and their study uses tools from PDE, Geometric Measure Theory, and Calculus of Variations. The goal of this talk is to present different free boundary problems, explain the main known results in this context, and give an overview of the current research and open problems. In particular, we will discuss a long-standing open question in the field which concerns the generic regularity of free boundaries, and present some new results in this direction.

Eugene Lytvynov (Swansea) - Orthogonal polynomials of Lévy white noise and umbral calculus

Consider the Gelfand triple $$\mathcal D\subset L^2(\mathbb R_+,dt)\subset \mathcal D'$$, where $$\mathcal D$$ is the space of all smooth functions on $$\mathbb R_+=[0,\infty)$$ with compact support and $$\mathcal D'$$ is the dual space of $$\mathcal D$$ with respect to the center space $$L^2(\mathbb R_+,dt)$$. Let $$\mu$$ be a Lévy white noise measure on $$\mathcal D'$$, i.e., a probability measure on $$\mathcal D'$$ such that $$X_t=\int_0^t \omega(t)\,dt$$ is a Lévy process. Thus, $$\omega(t)=\frac d{dt}X_t\in\mathcal D'$$ can be interpreted as a path of a Lévy white noise. We will introduce the notion of a polynomial sequence on $$\mathcal D'$$, and we will single out a class of Lévy white noises for which there exists an orthogonal polynomial sequence on $$\mathcal D'$$. This class includes Gaussian white noise, Poisson point process and gamma random measure. Each system of such orthogonal polynomials can be interpreted as a Sheffer sequence on $$\mathcal D'$$. The classical umbral calculus studies Sheffer polynomial sequences in the one-dimensional or multivariate setting. Extending Grabiner's result related to the one-dimensional umbral calculus, we will construct a class of spaces of entire functions on $$\mathcal D'_{\mathbb C}$$ — the complexification of $$\mathcal D$$ — spanned by Sheffer polynomial sequences on $$\mathcal D'$$. This will, in particular, extend the well-known characterization of the Hida test space of Gaussian white noise as a space of entire functions on $$\mathcal D_{\mathbb C}'$$.

Organisers

Jonathan Ben-Artzi (Cardiff)
Dmitri Finkelshtein (Swansea)