Applied Analysis and PDEs Seminar

June 25, 2021
Giorgio Fusco (Università degli Studi dell’Aquila, Italy)
On the existence of N-junctions for a symmetric nonnegative potential with N + 1 zeros
Abstract. We consider a nonnegative potential W : R²→R which is invariant under C_N, the rotation group of the regular polygon with N sides. We assume that {W=0}={0, a, ωa,..., ω^N-1a} , ω = rotation of ^2π⁄_N , for some a ∈ R² \ {0}. We prove that, if a certain condition is satified, there exists a N-junction, that is a solution U : R²→R² of the vector Allen-Cahn equation that avoids 0 and connects a, ωa, ..., ω^N−1a at infinity. The proof is variational and is based on sharp lower and upper bounds for the energy and on a new pointwise estimate for vector minimizers.

June 18, 2021
Valery Serov (University of Oulu, Finland)
Inverse scattering for quasilinear operators of higher order
Abstract. The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the multidimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator also. Moreover, in the absence of the uniqueness theorem, we present an approximate method (inverse backscattering Born approximation) for the reconstruction of the singularities of unknown coefficients.

June 11, 2021
Zhiyuan Geng (Basque Center for Applied Mathematics, Spain)
A free boundary problem on 2D liquid crystal droplets
Abstract. Liquid crystal droplets are of great interest from physics and applications. They are important in the studies of topological defects, both in a bulk and on a surface, in understanding of anisotropic surface energies and anchoring conditions of liquid crystals. In this lecture I will talk about a two-dimensional free boundary problem motivated by a liquid crystal droplet model. The problem occurs when droplets of nematic liquid crystals are dispersed in an isotropic polymeric medium. Determining the shape of the droplet and the equilibrium configuration of the liquid crystals leads to a shape optimization problem that minimizes both the elastic energy inside the droplet and the interfacial energy on the surface. I will present the existence result and describe several geometric properties of the free boundary. The relationship with the Weil-Petersson curve will also be discussed. This is a joint work with Fanghua Lin.

June 4, 2021
Fabrice Bethuel (Sorbonne Université, France)
Asymptotics for two-dimensional elliptic Allen-Cahn systems
Abstract. The formation of codimension-one interfaces for multi-well gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The vectorial case in contrast is quite open. This lack of results and insight is to a large extent related to the absence of known monotonicity formula. I will focus on the elliptic case in two dimensions, and presents some results which extend to the vectorial case in two dimensions most of the results obtained for the scalar case. I will also emphasize some specific features of the vectorial case.

May 28, 2021
Habib Ammari (Eidgenössische Technische Hochschule Zürich, Switzerland)
Wave interaction with subwavelength resonators
Abstract. In this lecture, the speaker will review mathematical and computational frameworks to elucidate physical mechanisms for manipulating waves in a robust way at scales beyond the diffraction limit using subwavelength structures. Imaging, sensing and spectroscopic techniques based on the use of subwavelength resonators, approaches which are rapidly gaining in popularity, allow one to overcome the severely ill-posed character of image reconstruction in biomedical applications and to achieve super-resolution imaging and sensing. The speaker will present the extraordinary properties of subwavelength resonators and construct a unified mathematical approach for modelling subwavelength confinement and guiding of waves as well as imaging and sensing using artificial materials. He will also evaluate the robustness of the proposed approaches with respect tο uncertainties in the geometrical or physical parameters.

May 21, 2021
Vladislav Kravchenko (Cinvestav, Mexico & Southern Federal University, Russia)
Direct and inverse Sturm–Liouville problems: A method of solution
Abstract. I will present some recent developments in the theory and practice of direct and inverse Sturm–Liouville problems on finite and infinite intervals which result in a new approach for solving direct and inverse spectral and scattering problems [V. V. Kravchenko, Direct and inverse Sturm–Liouville problems: A method of solution, Frontiers in Mathematics, Birkhäuser, 2020]. It is based on the notion of transmutation (transformation) operators and their efficient construction. Analytical representations for Sturm–Liouville equations are derived in the form of functional series revealing interesting special features and lending themselves to direct and simple numerical solution of a wide variety of problems.

May 14, 2021
Nicholas Alikakos (National and Kapodistrian University of Athens, Greece)
Sharp lower bounds for the vector Allen–Cahn energy and qualitative properties of minimizers
Abstract. We study vector minimizers {u_ε} with energy J_Ω(u) = ∫_Ω (^ε⁄₂ |∇u| ² + ¹⁄_ε W(u)) dx, where W > 0 on R^m \ {a₁,..., a_N}, m ≥ 1, for a bounded domain Ω ⊂ R² with certain geometrical features and u = g_ε on ∂Ω. We derive a sharp lower bound of J_Ω(u) as ε → 0, with two features: a) it involves half of the gradient and b) part of the domain Ω. Based on this we derive very precise (in ε) pointwise estimates up to the boundary for lim u_ε = u₀ as ε → 0. Depending on the geometry of Ω, u_ε exhibits either boundary layers or internal layers. We do not impose symmetry hypotheses and we do not employ Γ-convergence techniques. [Joint work with G. Fusco (L’Aquila)]

May 7, 2021
Michail Loulakis (National Technical University of Athens, Greece)
Optimal management of stochastic shallow lakes
Abstract. Lakes provide conflicting services as clear water resources and as waste sinks to industrial or agricultural activities. We study the welfare function of a control problem for the deposition of phosphorus in shallow lakes. Stochastic analysis provides crucial properties for the welfare function, e.g., regularity, monotonicity, boundary and asymptotic behaviour. We use such estimates to show that the welfare function is the unique constrained viscosity solution to a Hamilton–Jacobi–Bellman equation. Using the Barles–Souganidis scheme, we compute the welfare function numerically and we investigate the trajectories of the optimally controlled lake. [Joint work with G. Kossioris (Heraklion), A. Koutsibela (Athens), and P. Souganidis (Chicago)]

April 30, 2021
Vassilis Rothos (Aristotle University of Thessaloniki, Greece)
Localized structures in nonlocal NLS-type systems
Abstract. In the first part, we study the existence and bifurcation results for quasi periodic traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions and with polynomial type potentials. We employ variational ana topological methods to prove the existence of traveling waves in nonlocal DNLS lattice. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential (nonlocal NLS). While in the case of cubic and quintic interactions of the same kind (e.g. both attractive or both repulsive), only a symmetry breaking bifurcation can be identified, a remarkable effect that emerges e.g. in the setting of repulsive cubic but attractive quintic interactions is a “symmetry restoring” bifurcation. Namely, in addition to the supercritical pitchfork that leads to a spontaneous symmetry breaking of the anti-symmetric state, there is a subcritical pitchfork that eventually reunites the asymmetric daughter branch with the anti-symmetric parent one. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. The model is argued to be of physical relevance, especially so in the context of optical thermal media.

April 23, 2021
Giovanni Bellettini (Università degli Studi di Siena & ICTP Trieste, Italy)
Nonlocality of the area functional for two-codimensional graphs of nonsmooth maps in R⁴
Abstract. We discuss some properties of the area functional for graphs of certain nonsmooth maps u, in dimension two and codimension two. Geometrically, the problem is to understand how to fill the “holes” in the graph of u (which are due to its nonsmoothness) with smooth two-dimensional graphs in R⁴, keeping the area as small as possible. We focus attention on the vortex map u and, possibly, on particular piecewise constant maps.

April 16, 2021
Alexandros Eskenazis (University of Cambridge, U.K.)
The logarithmic Brunn–Minkowski conjecture
Abstract. We shall discuss the conjectured logarithmic Brunn–Minkowski inequality of Böröczky, Lutwak, Yang and Zhang (2012), which is a far-reaching refinement of the classical Brunn–Minkowski inequality for symmetric convex sets. After a quick recap of known special cases, we will explain an equivalent local form of the conjecture which is a functional inequality for functions defined on the boundary of symmetric convex sets. Time permitting, we will then show a proof (from joint work with G. Moschidis) of the Gardner–Zvavitch conjecture, which is formally weaker than the log-Brunn–Minkowski inequality.

April 9, 2021
Giulio Ciraolo (Università degli Studi di Milano, Italy)
On the shape of almost constant mean curvature hypersufaces
Abstract. Alexandrov’s theorem asserts that spheres are the only closed embedded hypersurfaces with constant mean curvature in the Euclidean space. In this talk we will discuss some quantitative versions of Alexandrov’s theorem. In particular, we will consider a hypersurface with mean curvature close to a constant and quantitatively describe its proximity to a sphere or to a collection of tangent spheres of equal radii in terms of the oscillation of the mean curvature. We will also discuss these issues for the nonlocal mean curvature, by showing a remarkable rigidity property of the nonlocal problem which prevents bubbling phenomena and proving the proximity to a single sphere.

April 2, 2021
Vassilis Papanicolaou (National Technical University of Athens, Greece)
A binary search scheme for determining all contaminated specimens
Abstract. Specimens are collected from N different sources. Each specimen has probability p of being contaminated (e.g., by SARS-CoV-2, in which case p is the prevalence rate), independently of the other specimens. Suppose we can apply group testing, namely take small portions from several specimens, mix them together, and test the mixture for contamination, so that if the test turns positive, then at least one of the samples in the mixture is contaminated.
 In this work we give a detailed probabilistic analysis of a binary search scheme for determining all contaminated specimens. More precisely, we study the number T(N) of tests required in order to find all the contaminated specimens, if this search scheme is applied. We derive recursive and, in some cases, explicit formulas for the expectation, the variance, and the characteristic function of T(N). Also, we determine the asymptotic behavior of the moments of T(N) as N → ∞ and from that we obtain the limiting distribution of T(N) (appropriately normalized), which turns out to be normal.

March 26, 2021
Vitali Vougalter (University of Toronto, Canada)
Solvability of some integro-differential equations with anomalous diffusion and transport
Abstract. The work deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions. [Joint work with V. Volpert (Lyon)]

March 12, 2021
Emmanuil Georgoulis (University of Leicester, U.K. & National Technical University of Athens, Greece)
Hypocoercivity-preserving Galerkin discretisations
Abstract. Degenerate differential evolution PDE problems are often characterised by the explicit presence of diffusion/dissipation in some of the spatial directions only, yet may still admit decay properties to some long time equilibrium. Classical examples include the inhomogeneous Fokker–Planck equation, Boltzmann equation with various collision kernels, systems of equation arising in micromagnetism or flow vorticity modelling, etc. In the celebrated AMS memoir Hypocoercivity, Villani introduced the concept of hypocoercivity to describe a framework able to explain decay to equilibrium in the presence of dissipation in some directions only. The key technical idea involved is to exploit certain commutators to overcome the degeneracy of dissipation.
 I shall present some results and ideas on the development of numerical methods which preserve the hypocoercivity property upon discretisation. As a result, such numerical methods will be suitable for arbitrarily long-time simulations of complex phenomena modelled by kinetic-type formulations. This will be achieved by addressing the key challenge of lack of commutativity between differentiation and discretisation in the context of mesh-based Galerkin-type numerical methods via the use of carefully constructed non-conforming weak formulations of the underlying evolution problems.

March 5, 2021
Gerassimos Barbatis (National and Kapodistrian University of Athens, Greece)
Best Sobolev constants in the presence of sharp Hardy terms
Abstract. We compute the best Sobolev constants for various Hardy–Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.

February 26, 2021
Ioannis Stratis (National and Kapodistrian University of Athens, Greece)
On an interior Calderón operator and a related Steklov eigenproblem for Maxwell’s equations
Abstract. We discuss a Steklov-type problem for Maxwell’s equations which is related to an interior Calderón operator and an appropriate Dirichlet-to-Neumann type map. The corresponding Neumann-to-Dirichlet map turns out to be compact and this furnishes a Fourier basis of Steklov eigenfunctions for the associated energy spaces. We provide natural spectral representations for the appropriate trace spaces, for the Calderón operator itself, and for the solutions of the corresponding boundary value problems subject to electric, or magnetic, boundary conditions on a cavity. [Joint work with P. D. Lamberti (Padova)]

February 19, 2021
Athanasios Yannacopoulos (Athens University of Economics and Business, Greece)
Hamilton–Jacobi–Bellman–Isaacs equations in Hilbert spaces with applications in decision making under uncertainty
Abstract. We present some recent results related to robust control for parabolic stochastic partial differential equations subject to model uncertainty. We address the problem in terms of a stochastic differential game which is subsequently treated by dynamic programming techniques. The value function is characterized in terms of a Hamilton–Jacobi–Bellman–Isaacs equation on a Hilbert space. The existence of mild solutions is shown, under conditions on the uncertainty set, and their use in the construction of game strategies is studied. As an application we illustrate our results on a spatially extended model in resource management. [Joint work with I. Baltas (Chios) and A. Xepapadeas (Athens/Bologna)]

February 12, 2021
Filippo Santambrogio (Université Claude-Bernard Lyon 1, France)
Lipschitz estimates on the time-discretization of the Fokker–Planck equation as a gradient flow in the Wasserstein space
Abstract. From the work by Jordan, Kinderlehrer and Otto it is known that some parabolic PDEs have a variational structure (of steepest descent type) in the Wassestein space of probability densities endowed with a distance coming from optimal transport. The most typical example is the Fokker–Planck equation ∂_tρ = Δρ + ∇·(ρ∇V), associated with the energy F(ρ) := ∫ ρlogρ + ρV. Due to this variational structure, a very natural time-discretization scheme can be built, known as the JKO scheme: at each time step the sum of F plus a suitable transport cost from the previous density is minimized, thus obtaining a recursive sequence of densities which approximate the solution of the PDE. It is interesting to see which bounds and regularity properties known to be satisfied by the solutions of the continuous-time equation are also satisfied in the discrete scheme.
 In the present talk, after recalling the main ingredients to understand the JKO scheme, I will give an estimate on the gradient of the solution and explain its interest. Obtaining it from the discrete scheme amounts to some easy manipulations on the Monge–Ampère equation. [Joint work with V. Ferrari (Paris)]

Last modified on June 23, 2021