Department of Mathematics |

*The seminar is taking place either in room A32 or in room A31. All talks are on Friday at 3:15 p.m. EEST (Athens).
*

**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} Δ^{2} with
| ∇ a(x) | ≤ c a(x)^{1⁄2},
deducing generation results for
A :=- (1+|x|)^{2a} Δ^{2}
in the case 0≤ a ≤ 2.
Morevoer, we will consider the Schrödinger
type operator
A =(1+|x|^{a})^{2} Δ^{2} +|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 **R**^{n}-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 **R**^{n} 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.