E 19 / Mathematikggebäude

Matthias Hofmann (Hagen): Spectral minimal partitions on unbounded graphs and domains We introduce spectral minimal partitions and their relations to eigenvalue problems. Recently, we proved existence and non-existence of spectral minimal partitions on unbounded metric graphs, where the operator considered on each of the partition elements is a Schrödinger operator of the form with suitable (electric) potential , which is taken as a fixed, underlying “landscape”. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum of the essential spectrum of the corresponding Schrödinger operator on the other, which recalls a similar principle for the eigenvalues of the latter: for any k∈N, the infimal energy among all admissible k-partitions is bounded from above by , and if it is strictly below , then a spectral minimal -partition exists. We illustrate our results with several examples of existence and nonexistence of minimal partitions. We conclude our talk with some recent ideas on generalizations on domains. Philippe Jaming (Université de Bordeaux): Null-controllability of the Generalized Baouendi-Grushin heat like equations This talk is devoted to null-controllability results for the heat equation associated to fractional Baouendi-Grushin operators where is a potential that satisfies some power growth conditions and the set is thick in some sense. This extends previously known results for potentials . The proof is based on a precised quantitative form of Zhu-Zhuge's spectral inequality for Schrödinger operators with power growth potentials. This is joint work with my student Yunlei Wang. Daniel Rosen (Dortmund): Randomized control of the heat equation We consider the problem of control of the heat equation on bounded cubes with random control set. Based on joint work in progress with I. Veselić. Jonathan Mui (Wuppertal): Heat equations with non-local Robin boundary conditions This talk concerns general results on diffusion equations, associated to uniformly elliptic operators with bounded real coefficients, on bounded Lipschitz domains with a simple type of non-local Robin boundary condition. It is of particular interest that, unlike the classical case of local boundary conditions, the solution semigroup in this case need not be positivity preserving. Nevertheless, we give conditions on the boundary operator for the semigroup to be ultracontractive, and for the generator to admit a positive leading eigenfunction (which are familiar properties in the classical case). These properties allow us to deduce the perhaps surprising conclusion that the semigroups are eventually positive. This is joint work with Jochen Glück.

Matthias Hofmann, Phillippe Jaming, Daniel Rosen, Jonathan Mui

Hagen, Université de Bordeaux, Dortmund, Wuppertal