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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: TBA
  
  Abstract: TBA
• 03.06.2025: Lauriane Chomaz (Universität Heidelberg).
  
  Title: TBA
  
  Abstract: TBA
  
• 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.
  
• 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