Oberseminar Analysis Winter 2024/25

Motivated by engineering and Photonics research on resonators in random or uncertain environments, we introduce rigorous randomizations of boundary conditions for acoustic wave equations in Lipschitz domains. First, a parametrization of essentially all m-dissipative boundary conditions in the boundary L2-space is constructed with the use of boundary tuples. Randomizations of these boundary conditions lead to acoustic operators random in the resolvent sense. We prove this using Neumann-to-Dirichlet maps and generalized Krein resolvent formulae. In order to pass to point processes of random eigenvalues, we give a description of random m-dissipative boundary conditions that produce acoustic operators with almost surely (a.s.) compact resolvents, and so, also with a.s. discrete spectra. Based on these results, examples of mathematically convenient randomizations are constructed in terms of eigenfunctions of the Laplace-Beltrami operator on the boundary. We show that for these special randomizations the resolvent compactness is connected with the Weyl law for the Laplace-Beltrami eigenvalues, and also discuss available Weyl-asymptotic results for the case of nonsmooth boundaries.

In the last decades, since the first experimental realizations of Bose-Einstein condensates, the study of large bosonic systems has been a very active field of research both in physics and in mathematics. In experiments Bose gases are often very dilute and can be well described in the Gross-Pitaevskii limit, i.e. as quantum systems of N confined particles, interacting through a potential with scattering length of order 1/N where N tends to infinity. In this talk we discuss a result on a hard-sphere Bose gas in the Gross-Pitaevskii regime. Namely, we prove a second order upper bound on the ground state energy matching the known expression of the energy for integrable potentials. We also discuss a new upper bound for hard-spheres in the thermodynamic limit where the number of particles and the size of the box are sent to infinity keeping the density fixed. Our result resolves the ground state energy up to an error of the order of the so-called Lee-Huang-Yang correction. Based on joint works with S.Cenatiempo, A.Giuliani, A.Olgiati, G.Pasqualetti, B. Schlein.

Telomeres are repetitive sequences situated at both ends of the chromosomes of eukaryotic cells. At each cell division, they are eroded until they reach a critical length that triggers a state in which the cell stops to divide: the senescent state. In this work, we are interested in the link between the initial distribution of telomere lengths and the distribution of senescence times. We propose a method to retrieve the initial distribution of telomere lengths, using only measurements of senescence times. Our approach relies on approximating our model with a transport equation, which provides a natural estimator for the initial telomere length distribution. We investigate this method from a theoretical point of view by providing bounds on the estimation error, pointwise and in all Lebesgue spaces. We also illustrate it with estimations on simulations, and discuss about its limitations related to the curse of dimensionality.

The gravity-Stokes system serves as a fundamental model for understanding the dynamics of incompressible fluids in certain regimes. We focus on the scenario where two fluids of different densities interact in a two-dimensional region without mixing. The density difference together with the gravity influence induce the dynamics of the two fluids and hence the evolution of the free interface arising between them. Through a contour dynamics approach, we address questions such as the existence of global solutions for this system and their asymptotic behavior, making emphasis on the properties of the free boundary. This talk is based in joint work with F. Gancedo and R. Granero Belinchón.

We consider the following boundary value problem on a compact Riemann surface split into two pieces by a collection of Jordan curves. Given an $L^2$ harmonic one-form on one of the pieces, we seek a harmonic one-form with the same boundary values and specified cohomology, which we call the "overfare" of the original form. When the Jordan curves are quasicircles, which are irregular curves arising in (for example) complex dynamics and Teichmüller theory, overfare is bounded. Furthermore, an associated scattering matrix is unitary. This has applications to approximation theory in complex analysis, Teichmüller theory, and conformal field theory. We give a non-technical overview of the analytic and geometric properties of overfare and related integral operators.

We will discuss socio-economic models based on interacting particle systems and their port-Hamiltonian structure. This structure is exploited to study long-time behaviour of dissipative systems with potential interactions. Then we turn our attention to optimal control tasks for interacting particle systems and discuss the derivation of the first-order optimality system in Wasserstein-sense. The theoretical findings are underlined with numerical results

In this talk, we investigate the global-in-time regularity of the two-dimensional incompressible Navier-Stokes equations in the presence of large variations in the viscosity coefficient. Through the analysis of two good unknowns, which are associated to the viscous stress tensor globally resp. locally, we establish the Lipschitz continuity of the velocity field, even when its normal derivative exhibits discontinuities across viscosity-interfaces. Specifically, we demonstrate that the two-phase two-dimensional Navier-Stokes equations admit a unique global-in-time solution that maintains the regularity of the free interface. This is joint work with Rebekka Zimmermann (KIT).

The inverse problem of Calderón, in its geometric formulation, asks if a Riemannian metric in a domain is determined up to isometry by boundary measurements of harmonic functions. Physically this corresponds to determining a matrix electrical conductivity function from voltage and current measurements on the boundary. This problem is open in general. In this talk we will discuss the hyperbolic analogue of the Calderón problem for the (Lorentzian) wave equation. We will show a rigidity result stating that any globally hyperbolic Lorentzian metric can be distinguished from the Minkowski metric. The result is valid for formally determined data and the method is based on distorted plane waves and geometric, topological and unique continuation arguments. This is joint work with L. Oksanen (Helsinki) and Rakesh (Delaware).