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Oberseminar: Calculus of Variations and Applications

Organizers: Phan Thành Nam, Arnaud Triay

Past semesters

• Winter semester 24
• Summer semester 24
• Winter semester 23
• Summer semester 23
• Winter semester 22

Summer Semester 2025

• 30.04.2025: Firoj Sk (University of Oldenburg).
  
  Title: On logarithmic p-Laplacian
  
  Abstract: In this talk, we introduce and analyze the logarithmic p-Laplacian \(L_{\Delta_p}\), a nonlocal operator of logarithmic order that arises as the formal derivative of the fractional p-Laplacian \((-\Delta_p)^s\) at \(s=0\). This operator serves as a nonlinear extension of the recently developed logarithmic Laplacian operator. We present a variational framework to study Dirichlet problems involving \(L_{\Delta_p}\) in bounded domains, which enables us to explore the relationship between the first Dirichlet eigenvalue and eigenfunction of both the fractional p-Laplacian and the logarithmic p-Laplacian. As a key result, we obtain a Faber–Krahn-type inequality for the first Dirichlet eigenvalue of \(L_{\Delta_p}\). Additionally, we examine the validity of maximum and comparison principles, showing that these depend on the sign of the first Dirichlet eigenvalue of \(L_{\Delta_p}\). Finally, we discuss a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic p-Laplacian. The talk is based on joint work with B. Dyda and S. Jarohs.
• 14.5.2025: Tobias Ried (Georgia Tech).
  
  Title: Domain branching in ferromagnets: elliptic regularity in action
  
  Abstract: The Landau-Lifshitz model of micromagnetics is a powerful continuum theory that describes the occurrence of magnetization patterns in a ferromagnetic body. In this talk I will discuss domain branching in strongly uniaxial materials resulting from the competition between a short-range attractive interaction (surface energy), a long-range repulsive interaction (stray field energy), and a non-convex constraint coming from the strong uniaxiality. On a mathematical level, we use modern tools from elliptic regularity theory, convex duality, ideas from statistical physics, and fine geometric constructions to describe the occurrence of domain branching through local energy estimates at the boundary of the sample (where the branching is infinitely fine). Our approach provides a robust framework for other domain branching problems and is the first step to prove self-similarity in a statistical sense. (Joint work with Carlos Román)
• 28.5.2025: Jiahao Wu (Peking University).
  
  Title: The 2nd order 2D behaviour of 3D Bose gases in the Gross-Pitaevskii regime.
  
  Abstract: In this talk we focus on a 3D interacting Bose gas strongly confined in one direction as motivated by experiments. We derive the 2nd order ground state energy formula for such system in the G-P limit. The interplay between parameters leads to three different finer regimes. Our work on one the hand is compatible with the classical 3D Lee-Huang-Yang formula, and on the other hand reveals the mechanism of smooth transition from 3D to the quasi 2D regimes, which was previously thought of as containing a jump. Our proof contains the classical 3D renormalizations and expected quasi-2D renormalizations, and the new dimensional coupling renormalizations. This talk is based on a joint work with Xuwen Chen and Zhifei Zhang (preprint: arXiv: 2401.15540).
• 03.06.2025: Lauriane Chomaz (Universität Heidelberg).
  
  Title: Stabilization by quantum fluctuations in ultracold gases of magnetic atoms : experimental observations and theory descriptions
  
  Abstract: Thanks to their high degree of control and tunability, ultracold atomic gases provide a rich platform for the study of quantum many-body effects. Ultracold gases of highly magnetic atoms exhibit unique interaction properties that lead to striking behaviors, both at the mean-field level and beyond [1]. A decade ago, a universal stabilization mechanism driven by quantum fluctuations was discovered in these gases. This mechanism prevents the systems from collapsing when the mean-field interactions become attractive, and instead allows exotic states of matter to arise, including ultradilute quantum droplets, crystallized quantum states, and especially the so-called supersolids [2]. In my colloquium, I will present the seminal observations of these states and how they emerged from the long-standing progress in the field. I will discuss the theoretical description of these systems via an effective mean-field theory, including the effect of quantum fluctuations via a higher-order effective interaction. I will outline our current understanding of the properties of these states and highlight open questions.
  [1] L. Chomaz & al, Dipolar physics: a review of experiments with magnetic quantum gases, Reports on Progress in Physics 86, 026401 (2023).
  [2] L. Chomaz, Quantum-stabilized states in magnetic dipolar quantum gases, arXiv preprint 2504.06221 (2025)
  
• 25.06.2025: Nikolai Leopold (Constructor University Bremen).
  
  Title: Derivation of the Maxwell–Schr¨odinger and Vlasov–Maxwell Equations from Non-Relativistic QED
  
  Abstract: This talk explores how Maxwell’s equations arise as an effective description of non-relativistic QED. In the first part, I will discuss the scaling regimes in which the quantized electromagnetic field is expected to behave classically. The second part focuses on the spinless Pauli–Fierz Hamiltonian in a semiclassical mean-field limit with many fermions. I will present recent results showing that, in the trace norm topology of reduced density matrices and in the large-particle-number limit, the system’s dynamics can be approximated by a fermionic variant of the Maxwell–Schr¨odinger equations as well as by the non-relativistic Vlasov–Maxwell system for extended charges. This talk is based on the preprints arXiv:2411.07085 and arXiv:2308.16074, the latter being a joint work with Chiara Saffirio.
  
• 30.06.2025: Daniel Grieser (Carl von Ossietzky Universität Oldenburg).
  
  Title: The Dirichlet-Neumann operator for fibred cusp geometries
  
  Abstract: We consider Riemannian manifolds with boundary where the boundary exhibits singularities of fibred cusp type, or are conformal to these. A simple example is the complement of two touching balls in $\mathbb{R}^n$. This type of singularity (at the touching point), in case $n=2$, is often called an incomplete cusp (or horn). Other examples, conformal to these types of spaces and with 'singularity' at infinity, are fundamental domains of Fuchsian groups and uniformly fattened infinite cones in $\mathbb{R}^n$. The Dirichlet-Neumann (DN) operator on a Riemannian manifold with boundary maps Dirichlet boundary data of harmonic functions to their Neumann data. This operator is well studied in the smooth compact case, for example it is known that it is a pseudodifferential operator (PsiDO), and its spectrum (the Steklov eigenvalues) has been studied intensively, as well as the inverse problem for it.
  We show that the DN operator for fibred cusp singularities are in a PsiDO calculus adapted to the geometry, the so-called phi-calculus. This yields a precise description of their integral kernels near the singularities. In the talk I will introduce the necessary background on the phi-calculus, and also discuss some of the spectral properties of the DN operator in this setting.
  This is joint work with K. Fritzsch und E. Schrohe.
  
• 02.07.2025: Marco Olivieri (University of Copenhagen).
  
  Title: TBA
  
  Abstract: TBA
• 09.07.2025: Fabrizio Caragiulo (SISSA Trieste).
  
  Title: TBA
  
  Abstract: TBA
• 16.07.2025: Florian Haberberger (LMU Munich).
  
  Title: TBA
  
  Abstract: TBA