Applied Analysis and PDEs Seminar

November 24, 2023, and December 1, 2023
Jacopo Schino (North Carolina State University)
Mini course: A novel and simple approach to normalised solutions to Schrödinger equations ( Abstract).

March 29, 2024
Vitali Vougalter (University of Toronto)
Solvability of some integro-differential equations with the double scale anomalous diffusion in higher dimensions.
Abstract. The article is devoted to the studies of the existence of solutions of an integro-differential equation in the case of the double scale anomalous diffusion with the sum of the two negative Laplacians raised to two distinct fractional powers in R^d, d=4,5. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for the non-Fredholm elliptic operators in unbounded domains are used.
Webex link (3:00 PM - 4:00 PM, Friday, March 29, 2024) Meeting password: 2Cr5tmRxzu4

April 5, 2024
Spyridon Filippas (University of Helsinki)
On unique continuation for waves in singular media.
Abstract. The problem of unique continuation consists in recovering the whole wave from a partial observation and has applications to control theory and inverse problems. After presenting some fundamental results of the theory we will explain how one can prove a logarithmic stability result for wave operators whose metric exhibits a jump discontinuity across an interface. We make no assumption about the sign of the jump or the geometry of the interface. The key ingredient of our proof is a local Carleman inequality near the interface. Using a propagation argument, we derive then a global stability estimate.

April 12, 2024
Orestis Vantzos (Vantzos Research SMPC, Athens, Greece)
Pattern formation in Ginzburg-Landau potentials with hard obstacles.
Abstract. We present a new class of vector-valued phase field models, where the values of the phase parameter are constrained to a convex set. Like the classic Ginzburg-Landau functional, these models favour functions that partition the domain into subdomains, where the function takes one of a number of distinct values corresponding to distinct phases, separated by interfaces of small thickness. We characterise the phases and interfaces of the proposed generalized Ginzburg-Landau functional, in particular with respect to their dependency on the geometry of the convex constraint set. Furthermore, we introduce an efficient proximal gradient solver to study numerically their L2- gradient flow, i.e. the associated generalized Allen-Cahn equation. We look at different choices for the shape of the convex constraint set, leading to the formation of a number of distinct patterns.

April 19, 2024
Nicholas Alikakos (University of Athens, EKPA, Greece)
Multi-phase Minimizers for the Allen-Cahn System on the plane.
Abstract. In this talk we investigate multi-phase minimizers for the Allen-Cahn system on the plane. Our emphasis in on distinct surface tension coefficients. The proofs do not rely on symmetry. Coexistence of an arbitrary number of phases is related to the existence of the relevant minimizing cones for the minimal partition problem. For example, the orthogonal cross with four phases is minimizing for certain class of surface tension coefficients. We focus on two examples: the entire solution for the triple junction, and a four-phase minimizer with three-phase Dirichlet data (the triangle). The results presented in the talk are based on joint work with Zhiyuan Geng (Triple Junction), and with Dimitrios Gazoulis (The Triangle).

April 26, 2024
Nikolaos Roidos (University of Patras)
The heat asymptotics near cones and the spectrum of the Laplacian on the cross-sections.
Abstract. We consider the heat equation on manifolds with isolated conical singularities and investigate the full asymptotic behaviour of the solutions near the conical tips in terms of the spectrum of the Laplacian on the cross-sections.
Webex link (3:00 PM - 4:00 PM, Friday, April 26, 2024) Meeting password: 2Cr5tmRxzu4

May 17, 2024
Eric Stachura (Kennesaw State University, USA)
Quantitative Analysis of passive intermodulation and surface roughness.
Abstract. I will discuss a basic model of passive intermodulation (PIM). PIM occurs when multiple signals are active in a passive device that exhibits a nonlinear response. It is known that certain nonlinearities (e.g. the electro-thermal effect) which are fundamental to electromagnetic wave interaction with matter should be accounted for. In this talk, I will discuss existence and regularity of solutions to a simple model for PIM when the underlying domain is Lipschitz. This in particular includes a temperature dependent conductivity in Maxwell's equations, which themselves are coupled to a nonlinear heat equation. I will also discuss challenges related to a similar problem when the permittivity ε depends on temperature. This is joint work with Niklas Wellander and Elena Cherkaev.
Webex link (3:00 PM - 4:00 PM, Friday, May 17, 2024) Meeting password: 2Cr5tmRxzu4

May 24, 2024
Francesco Ferraresso (University of Sassari, Italy)
Spectral analysis of dissipative Maxwell systems.
Abstract. Dissipative Maxwell systems find frequent application in the modelling of electromagnetic wave propagation through conductive media. In the presence of non-trivial conductivity, the medium absorbs part of the EM energy of the wave. From a mathematical point of view, conductivity makes the underlying Maxwell operator non-selfadjoint. I will discuss a few recent results regarding the essential spectrum and the spectral approximation of dissipative Maxwell systems in unbounded domains of the three dimensional Euclidean space. Under certain assumptions on the behaviour of the coefficients at infinity, the essential spectrum decomposes in two parts, one of which is non-empty even in bounded domains. Nevertheless, the eigenvalues of finite multiplicity of the system can be approximated exactly by means of the domain truncation method. We will then see how these results generalise to situations of relevance in applications, such as Drude-Lorentz metamaterials and semi-transparent Faraday layers. Based on joint work with S. Bögli (Durham), M. Marletta (Cardiff), and C. Tretter (Bern).
Webex link (3:00 PM - 4:00 PM, Friday, May 17, 2024) Meeting password: 2Cr5tmRxzu4

May 31, 2024
Federica Gregorio (University of Salerno, Italy)
Bi-Laplacians with unbounded coefficients.
Abstract. In this talk we will prove generation results in L^p(R^N) spaces and domain characterization of some fourth-order operators with polynomially growing coefficients. More specifically, we will focus on operators of the form A=-a(x)² Δ² with | ∇ a(x) | ≤ c a(x)^1⁄₂, deducing generation results for A :=- (1+|x|)^2a Δ² in the case 0≤ a ≤ 2. Morevoer, we will consider the Schrödinger type operator A =(1+|x|^a)² Δ² +|x|^2b in the case N ≥ 5, a, b>0,b >a-2.
Webex link (3:00 PM - 4:00 PM, Friday, May 17, 2024) Meeting password: 2Cr5tmRxzu4

June 7, 2024
Konstantinos Zemas (University of Bonn, Germany)
Stability aspects of the Möbius group of the sphere.
Abstract. In this talk I would like to discuss quantitative stability aspects of the class of Möbius transformations of the sphere among maps in the critical Sobolev space (with respect to the dimension). The case of sphere- and Rⁿ-valued maps will be addressed. In the latter, more flexible setting, unlike similar in flavour results for maps defined on domains, not only a conformal deficit is necessary, but also a deficit measuring the distortion of the sphere under the maps in consideration, which is introduced as an associated isoperimetric deficit. The talk will be based on previous works in collaboration with Stephan Luckhaus, Jonas Hirsch, and more recent ones with Andre Guerra and Xavier Lamy.

June 14, 2024
Dimitrios Gazoulis (University of Athens, EKPA, Greece)
A Pointwise Gradient bound for Pucci's equations and it's consequences
Abstract. We consider Pucci's extremal operators in Rⁿ and study entire solutions of the respective Pucci's equations. In this talk, we will prove a pointwise gradient estimate that generalizes the Modica inequality for fully nonlinear elliptic equations. This bound is sharp, in the sense that, when equality is attained at some point, then the level sets of the solutions will be hyperplanes. We will also provide some applications, such as Liouville theorems and Harnack-type inequalities.