This seminar is organized in the framework of the reasearch project *Sensitivity analysis of partial differential equations in the mathematical theory of electromagnetism* of the University of Padova. Special attention is devoted to time harmonic Maxwell equations, eigenvalue problems and their qualitative properties, as well as to the analysis of the associated function spaces.

Participation is mainly upon invitation. Interested people should contact the organizers by e-mail in order to receive instructions and the link to the Zoom session of interest.

**Forthcoming seminars:**

**April 13th 2022 at 17:30 **(Italian Time): Prof. Maria Rosaria Lancia (Università degli Studi di Roma "La Sapienza")

Title: *The magnetic properties of fractal structures*

Abstract: *Trying to understand the magnetic properties of fractal structures is a new challenge from both the practical and theoretical point of view. In this talk, I present some results for a magnetostatic problem in a 3D “cylindrical" domain of Koch type and in its corresponding prefractal approximating domains. Crucial tools, to prove well-posedness results, are a generalized Stokes formula and the Friedrichs inequality for domains with fractal boundary. We investigate the convergence of the pre-fractal solutions to the limit fractal one. We also consider the numerical approximation of the pre-fractal problems via FEM and we give a priori error estimates. Some numerical simulations are also shown.
This is joint work with S.Creo (Sapienza, Università di Roma), M.Hinz (Bielefeld University), A. Teplyaev (University of Connecticut).*

**April 6th 2022 at 17:30 **(Italian Time): Prof. Rainer Picard (TU Dresden, Germany)

Title: *Maxwell's Equation as an Abstract Dynamic Friedrichs System*

Abstract: *We shall inspect some aspects of Maxwell's equations by presenting them within the framework of abstract Friedrichs systems. Since we will be concerned with time-dependence, we are indeed considering more specifically abstract dynamic Friedrichs systems. In the abstract setting we shall characterize time-independent boundary conditions leading to maximal accretive spatial operators. The abstract setting is then illustrated by a more detailled investigation of Maxwell's equation in complex media. We conclude our presentation with some structural observations in connection with the so-called extended Maxwell system*.

**Past seminars:**

**January 26th 2022 at 17:30 **(Italian Time): Prof. Darko Volkov (Worcester Polytechnic Institute, USA)

Title: *A Well-Posed Surface Integral Equation for the Maxwell Dielectric Problem*

Abstract: *The free space Maxwell dielectric problem can be reduced to a system of surface integral equations (SIE). A numerical formulation for the Maxwell dielectric problem using an SIE system presents two key advantages: first, the radiation condition at infinity is exactly satisfied, and second, there is no need to artificially define a truncated domain. Consequently, these SIE systems have generated much interest in physics, electrical engineering, and mathematics, and many SIE formulations have been proposed over time. In this talk we introduce a new SIE formulation which is in the desirable operator form identity plus compact, is well-posed, and remains well-conditioned as the frequency tends to zero. The unknowns in the formulation are three dimensional vector fields on the boundary of the dielectric body. The resulting SIE discussed in this talk is derived from a formulation developed in earlier work. Our initial formulation utilized linear constraints to obtain a uniquely solvable system for all frequencies. The new SIE introduced and analyzed in this talk combines the integral equations from this initial formulation with new constraints. We show that the new system is in the operator form identity plus compact in a particular functional space, and we prove well-posedness at all frequencies and low-frequency stability of the new SIE.*

**June 23rd 2021 at 17.00** (Italian Time): Prof. Serge Nicaise (LAMAV, Université Polytechnique Hauts-de-France, Valenciennes, France)

Title: *Stability and asymptotic properties of a linearized hydrodynamic medium model for dispersive media in nanophotonics*

Abstract: *In this talk, we will present a linearized hydrodynamical model describing the response of nanometric dispersive metallic materials illuminated by optical light waves, a situation occurring in nanoplasmonics. The model corresponds to the coupling between the Maxwell system and a PDE describing the evolution of the polarization current of the electrons in the metal. First by using semigroup theory its well posedness will be shown. Then using a frequency approach, polynomial (and optimal) stability results for different boundary conditions will be presented.*

*This is a join work with Claire Scheid (Université Côte d'Azur, France).*

**June 1st 2021 at 17.00** (Italian Time): Prof. Rolando Magnanini (University of Florence, Italy)

Title: *The complex eikonal equation in 2D *

Abstract: * Theories of monochromatic high-frequency electromagnetic fields have been designed by Keller, Felsen, Kravtsov, Ludwig, and others to represent features that are ignored by geometrical optics. These theories make use of eikonals that encode information on both phase and amplitude — in other words, they are complex-valued. Any (real-valued) geometric optical eikonal, which describes conventional rays in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, is used. In this talk, I will give an account of these ideas in dimension 2. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, this theory entails a system of two first order PDEs, which decouples into two conjugate degenerate elliptic PDEs. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, functionals of the Calculus of Variations, and a Bäcklund transformation. *

**May 19th 2021 at 17.00** (Italian Time): Prof. Giovanni Franzina (IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone", Rome, Italy)

Title: *Electromagnetism in seismology *

Abstract: *We model magnetic anomalies at Earth’s surface due to hypogene co-seismic sources with classic magneto-quasistatic Maxwell’s equations in heterogeneous media. *

**May 12th 2021 at 17.00** (Italian Time): Prof. Marco Marletta (Cardiff University, United Kingdom)

Title: *An inverse problem in electromagnetism with partial data *

Abstract: * We consider the problem of detemining the permeability, permittivity and conductivity in a time-harmonic Maxwell system from measurements of the tangential components of the electric and magnetic fields, taken on an arbitrarily small, open subset of the boundary. We prove that under suitable hypotheses the problem has a unique solution. The key to the proof is a Runge-type theorem, which depends on a unique continuation principle. This problem arises from applications to Magnetic Induction Tomography (MIT) and our work was part of a joint project with Paul Ledger in Swansea School of Engineering (now in Keele). The results discussed in this seminar were funded by EPSRC grant EP/K024078/1 and published in a joint paper with Malcolm Brown and Juan Manuel Reyes Gonzales, https://doi.org/10.1016/j.jde.2016.01.002 *

**March 31st 2021 at 17.00** (Italian Time): Prof. Xavier Claeys (Université Pierre et Marie Curie Laboratoire Jacques-Louis Lions, Paris, France)

Title: *Radiation condition and instability phenomenon at a corner interface between a dielectric and a negative material*

Abstract: *In the present talk, we consider a 2D second order transmission problem involving a piece-wise constant coefficient in the principal part. This coefficient takes two values with opposite signs, which delimits an interface between two media. Although this problem may look elliptic at first glance, it is not, due to the sign change of the coeffcient. In the case where the interface contains a corner, which we will assume, the problem is not even Fredholm in a standard Sobolev setting if the contrast between the two media belongs to some critical interval. We will focus on this situation, showing that the problem recovers Fredholmness in a different functional setting, that relies on weighted Sobolev spaces and some outgoing radiation condition at the corner. We will also discuss recent results of asymptotic analysis that examine the effect of rounding the corner in such a context. We will present an exotic instability phenomenon appearing for our problem as the size for the rounded corner goes to zero.*

Slide presentation Prof. Claeys

**March 17th 2021 at 17.00** (Italian Time): Prof. Patrick Ciarlet (ENSTA Paris, France)

Title: *Time-harmonic electromagnetic waves in anisotropic media: theoretical and numerical analysis*

Abstract: * We study the time-harmonic Maxwell’s equations in a bounded domain, with electric permittivity and magnetic permeability that are complex, possibly non-hermitian, tensor fields. Each tensor field verifies an ellipticity condition. First, the well-posedness of the variational formulation corresponding to the problem with Dirichlet, respectively Neumann, boundary condition, is proven. Then, the regularity results of the electromagnetic fields are determined by using shift theorems for second-order divergence elliptic operators. Finally, the discretization with a H(curl)-conforming approximation based on edge finite elements is considered. The a priori error estimate is derived, and verified on elementary benchmarks.*

**Slide presentation Prof. Ciarlet**

**March 3rd 2021 at 17.00** (Italian Time): Prof. Fioralba Cakoni (Rutgers University, USA)

Title: *The Electromagnetic Transmission Eigenvalue Problem*

Abstract:* In this presentation we discuss the state-of-the-art of the transmission eigenvalue problem for Maxwell’s equations. The transmission eigenvalue problem is at the heart of inverse scattering theory for inhomogeneous media. It has a deceptively simple formulation but presents a perplexing mathematical structure, in particular it is a non-selfadjoint eigenvalue problem. Transmission eigenvalues are related to interrogating frequencies for which there is an incident field that doesn’t scatter. We will present some results and open problems on related spectral questions central to inverse scattering theory including: 1) discreteness of the spectrum that is closely related to the determination of the support of inhomogeneity from scattering data using non iterative methods, 2) location of transmission eigenvalues in the complex plane that is essential to the time domain inversion methods, 3) existence of transmission eigenvalues as well as the accurate determination of real transmission eigenvalues from scattering data, which has became important since real transmission eigenvalues can be used to obtain information about electromagnetic properties of the media.*

Slide presentation Prof. Cakoni

**February 3rd 2021 at 17.00** (Italian Time): Prof. Frank Hettlich (KIT - Karlsruhe Institute of Technology, Germany)

Title: *The Domain Derivative for Time Harmonic Electromagnetic Scattering Problems*

Abstract: *The dependence of time harmonic electromagnetic waves on the shape of scattering object is a challenging problem in inverse scattering theory as well as for shape optimization. We introduce the domain derivative for time harmonic electromagntic elds. By a variational approach existence and representations of such domain derivatives can be shown in case of different boundary conditions in appropriate Sobolev spaces. Furthermore, the appraoch is extended in showing a second derivative with respect to perturbations of the object. The characterizations of the domain derivatives can be applied to inverse scattering problems. As an illustration we consider the reconstruction of the shape of a perfectly conducting scatterer from the far field pattern of one scattered wave. It is shown that characterizations of the domain derivatives can be used succefully in iterative regularization schemes for such an severly ill-posed problem. Furthermore, also an application of the derivative for a shape optimization problem is discussed.*

Slide presentation Prof. Hettlich

**January 13th 2021 at 17.00** (Italian Time): Prof. Giovanni S. Alberti (University of Genoa, Italy)

Title: *Regularity theory for Maxwell’s equations *

Abstract: * In this talk, we will review the main results regarding the regularity properties of the solutions to time-harmonic Maxwell’s equations. The main tool is the Helmholtz decomposition of the electric and magnetic fields, which allows us to reduce Maxwell’s equations to a system of elliptic equations for the scalar and vector potentials. The regularity of the potentials then follows by classical elliptic regularity theory. We will discuss regularity in Sobolev and Hölder spaces, and consider interior as well as boundary estimates (in smooth domains). *

Slide presentation Prof. Alberti

**December 9th 2020 at 17.00** (Italian Time): Prof. Dirk Pauly (Duisburg Essen University, Germany)

Title: *FA-TOOLBOX: solving PDEs with Hilbert Complexes *

Slide presentation Prof. Pauly

**November 25th 2020 at 17.00** (Italian Time): Prof. Samuel E. Cogar (Rutgers University, USA)

Title: *Steklov eigenvalues for Maxwell’s equations*

Abstract: *A recent area of interest has been the development of new eigenvalue problems arising from scattering theory that may be used as target signatures in nondestructive testing of materials. In this talk I will focus on the Steklov eigenvalue problem for Maxwell’s equations. After motivating this area of research with the simpler Steklov eigenvalue problem for the Helmholtz equation, I will introduce the time-harmonic Maxwell’s equations and give some background on their study. The corresponding Steklov problem will then be described, including the modified versions that have been devised to correct some degenerate behavior in the eigenvalues. I will conclude with a further modification of my own for which infinitely many Steklov eigenvalues exist without any additional assumptions on the coefficients, a result that is not available for other classes of Steklov eigenvalues. *

Slide presentation Prof. Cogar