Oberseminar Analysis Winter 2026/27

Spontaneous pattern formation is classically explained by Turing's diffusion-driven instability (DDI), typically realised by a slowly diffusing activator and a fast-diffusing inhibitor. In this talk, I discuss two settings beyond this classical picture and ask, in each, which pattern is actually selected. First, I consider reaction-diffusion systems in which some components do not diffuse, so-called reaction-diffusion-ODE systems. Their stationary solutions can be less regular than in classical reaction-diffusion systems, exhibiting for example jump discontinuities, and the linearised operator has nontrivial essential spectrum. I derive sufficient conditions for DDI in systems with three spatial scales (non-diffusing, slow- and fast-diffusing components), and show that the destabilisation mechanism influences the stability of the emerging patterns. I also construct patterns with jump discontinuities, which exist largely independently of the diffusion coefficients. Deciding between regular and jump-discontinuous patterns requires a nonlinear stability theory for bounded, possibly discontinuous, steady states. I show that, in the space of bounded functions, a negative spectral bound of the linearisation implies nonlinear stability and a positive one implies instability, without requiring a spectral gap. Second, I present a mechanochemical model for head formation in the freshwater polyp Hydra, in which tissue mechanics takes the role of the fast-diffusing inhibitor. A one-dimensional reduction leads to a nonlocal reaction-diffusion equation, for which only single-peak patterns can be stable, consistent with the single body axis of Hydra. A bifurcation analysis reveals subcritical branches and bistability between the constant state and nonconstant stationary solutions, so that in this regime patterns emerge only from sufficiently large perturbations of the homogeneous state. Based on joint work with T. André, S. Cygan, C. Kowall, A. Marciniak-Czochra and D. Oelz.

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This talk is devoted to the characterization of removable singularities for Hölder continuous weak solutions to degenerate p-parabolic equations, with the model \[\partial_t u − div(︂ |∇u|^{p−2}∇u)︂ = 0 , p\ge 2 . (1)\] In more detail, we study the following problem: how large can the set E be so that the solution to the PDE outside of $E$, $\alpha$-Hölder continuous up to $E$, automatically is also a solution along $E$. In the simplest elliptic case $−∆u = 0$, it is known ([2]) that the set $E$ is removable for any of $u ∈ C^{0,α}$ if $H^{n−2+α}(E) = 0$. The problem was further studied for elliptic equations of the type $−∆pu = 0$ ([3]), and for parabolic equations built upon (1), with sufficient smoothness of the operator ([4]). I will show a new approach that allows characterizing removable sets of Hölder continuous solutions to $\partial_t u − divA(x, t, ∇u) = 0$ , where $A$ is any monotone operator of $p$-growth with respect to the gradient variable. Specifically, in the geometry intrinsic to the problem, removability of $E$ is again equivalent to $E$ having Hausdorff-type measure $0$. This is a joint work with Theo Elenius, Leah Kurki Schätzler, and David Stolnicki ([1]).

Literature

[1] M. Borowski, T. Elenius, L. Schätzler, and D. Stolnicki. Carleson-type removability for p-parabolic equations. Dec. 2025. , to appear in Trans. Amer. Math. Soc.

[2] L. Carleson. Removable singularities of continuous harmonic functions in Rm. Math. Scand., 12:15–18, 1963.

[3] T. Kilpeläinen and X. Zhong. Removable sets for continuous solutions of quasilinear elliptic equations. Proc. Amer. Math. Soc., 130(6):1681–1688, 2002.

[4] T. Kuusi, G. Mingione, and K. Nyström. Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets. J. Math. Pures Appl., 101(2):119–151, Feb. 2014.

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In this talk, we study an incompatible two-well problem that exhibits some surprising flexibility properties. We consider two variants of the problem: one without gauge invariance and one with SO(2) invariance. Upon taking a glimpse at the relaxations of the energies, we discuss a connection between these variational problems and the theory of quadrature domains. This link allows us to construct minimizers with finite surface energy via tools from potential theory and free-boundary problems, including partial balayage, mother bodies and a squeezing-type Hele-Shaw flow. The talk is based on ongoing joint work with Lennart Machill and Angkana Rüland.

We present a variational definition of the essential spectrum suitable for nonlinear operators such as the Dirichlet p-Laplacian. We then provide a geometric characterisation of the bottom of the essential spectrum by extending Persson's theorem to this framework. Finally, we show that the Ljusternik-Schnirelmann minmax levels of the constrained p-Dirichlet integral are attained whenever one of these levels is below Persson's threshold. This is based on joint work with Lorenzo Brasco, Giovanni Franzina, and Francesca Prinari.