Oberseminar Analysis Winter 2025/26

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. 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 case a sharp functional inequality admits optimizers, we are interested in improving the inequality by adding terms that involve a distance to the set of optimizers. Such refinements are known as (quantitative) stability results. In this talk, I first provide a short introduction to the topic of stability of functional inequalities. Following this, I present the $\sigma_2$-curvature inequality, a variational characterization of a fully nonlinear Yamabe-type equation, and explain how stability of this inequality can be established. As we will see, in contrast to previous Hilbert-space results, the distance to the set of optimizers is measured naturally in terms of two different Sobolev norms, for which optimal exponents are provided. Finally, I describe how the presented methods can be applied to improve an inequality by De Lellis and Topping, which in turn is a refinement of a well-known rigidity result by Schur. The talk is based on two joint works with Rupert Frank and Tobias König, respectively.

In 1990, Hoffman and Meeks proved the strong halfspace theorem for minimal surfaces, that is any connected, proper, possibly branched minimal surface in three-dimensional Euclidean space that is contained in a halfspace must be a plane. In a joint work with Matteo Cozzi, we prove the strong halfspace theorem for nonlocal minimal surfaces. Interestingly, our result holds for hypersurfaces of any dimension, in direct contrast to the classical case which does not hold for hypersurfaces of dimension three or higher. In the first half of my talk, I will introduce and motivate nonlocal minimal surfaces and try to highlight some interesting similarities/differences with their classical counterparts. In the second half, I will discuss the nonlocal halfspace theorem including some ideas of the proof.

We associate to every function $u\in GBD(\Omega)$ a bounded Radon measure $\mu_u$ with values in the space of symmetric matrices, which generalises the distributional symmetric gradient $Eu$ defined for functions of bounded deformation. We show that this measure $\mu_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures $\mu^a_u$, $\mu^c_u$, and $\mu^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(\Omega)$ functions. We then characterise the space $GSBD(\Omega)$, originally defined only by slicing, as the space of functions $u\in GBD(\Omega)$ such that $\mu^c_u=0$. This is joint work with Gianni Dal Maso.

Onsager's conjecture states that 1/3 is the critical spatial (Hölder) regularity threshold for energy conservation by weak solutions of the incompressible Euler equations. We consider Onsager’s conjecture for the Euler equations in the case of a bounded domain, as boundary effects play a crucial role in hydrodynamic turbulence. We present a regularity result for the pressure in the Euler equations, which is fundamental for the proof of the conservation part of the Onsager conjecture (in the presence of boundaries). As an essential part of the proof, we introduce a new weaker notion of boundary condition which we show to be necessary by means of an explicit example. Moreover, we derive this new boundary condition rigorously from the weak formulation of the Euler equations. In addition, we consider an analogue of Onsager’s conjecture for the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a ‘family’ of Onsager conjectures for these equations. Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. If time allows, I will also present some recent results on a model from sea-ice dynamics, which is the elastic-viscous-plastic sea-ice model. These are joint works with Claude Bardos, Xin Liu, Simon Markfelder, Marita Thomas and Edriss S. Titi.

We consider a class of dynamical systems that are described by time and space dependent partial differential equations. This class fits perfectly the port-Hamiltonian framework. We cover the wave equation, Maxwell's equations, the Kirchhoff-Love plate model, piezo-electromagnetic systems and many more. Our goal is to characterize boundary conditions that make the systems passive (the energy of solutions decays). This is done by constructing a boundary triple for the underlying differential operator. As a by-product we develop the theory of quasi Gelfand triples, which enables us to regard L2 boundary conditions even though the "natural" boundary spaces are neither included nor covering L2.