Ansprechpartner: Birgit Jacob und Balint Farkas (Wuppertal), Delio Mugnolo (Hagen), Ivan Veselić (Dortmund)
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 $-\Delta + V$ with suitable (electric) potential $V$, 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 $\lambda_{ess}$ 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 $\lambda_{ess}$, and if it is strictly below $\lambda_{ess}$, then a spectral minimal $k$-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.
This talk is devoted to null-controllability results for the heat equation associated to fractional Baouendi-Grushin operators $$ \partial_t u+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u=\mathbf{1}_\Omega h $$ where $V$ is a potential that satisfies some power growth conditions and the set $\Omega$ is thick in some sense. This extends previously known results for potentials $V(x)=|x|^{2k}$. 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.
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.
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.
Um den Zoom Zugangscode zu erhalten, können Sie eine Email an Ivan Veselic schreiben oder ihn auf der Moodle-Seite der TU Dortmund bzw. der Universitätsallianz Ruhr einsehen.
Eine Folge \((z_n)\) in der Einheitskreisscheibe ist eine interpolierende Folge für \(H^\infty\), falls es zu jeder beschränkten Folge \((w_n)\) eine beschränkte holomorphe Funktion \(f\) auf der Einheitskreisscheibe mit \(f(z_n) = w_n\) für alle \(n\) gibt. Diese Folgen wurden von Lennart Carleson charakterisiert. Ich werde über eine Verallgemeinerung von Carlesons Satz auf andere Klassen von Funktionen reden. Diese Verallgemeinerung nutzt die Lösung des Kadison-Singer-Problems von Marcus, Spielman und Srivastava. Der Vortrag beruht auf einer gemeinsamen Arbeit mit Alexandru Aleman, John McCarthy und Stefan Richter.
We report on recent results concerning partial differential operators with constant coefficients \(P(\partial)\) for which the characteristic set \(\{\xi\in\mathbb{R}^d;\,P_m(\xi)=0\}\) of its symbol \(P\in\mathbb{C}[X_1,\ldots,X_d]\) is a one-dimensional subspace of \(\mathbb{R}^d\). Here \(P_m\) denotes the principal part of \(P\), i.e. \(P_m(\xi):=\sum_{|\alpha|=m}a_\alpha \xi^\alpha\) for \(P(\xi)=\sum_{|\alpha|\leq m}a_\alpha\xi^\alpha\) with minimal \(m\in\mathbb{N}_0\). Among others, this class of partial differential operators contains the time-dependent free Schrödinger operator as well as non-degenerate parabolic operators like the heat operator. We characterize those open subsets \(X\) of \(\mathbb{R}^d\) for which \(P(\partial)\) is surjective on \(C^\infty(X)\), or equivalently on \(\mathscr{D}_F'(X)\), the space of distributions of finite order on \(X\). Moreover, we give a sufficient geometrical/topological condition for pairs of open subsets \(X_1\subseteq X_2\) of \(\mathbb{R}^d\) to be \(P\)-Runge pairs, which means that every smooth solution, resp. distributional solution, of the equation \(P(\partial)u=0\) in \(X_1\) can be approximated by smooth solutions, resp. distributional solutions, of the same equation in \(X_2\). This condition is in the spirit of Runge's Approximation Theorem from complex analysis which deals with the case when \(P(\partial)\) is the Cauchy-Riemann operator. Finally, we show that under the additional assumption of semi-ellipticity for such a differential operator surjectivity on \(C^\infty(X)\) implies that its kernel \(C_P^\infty(X)=\{f\in C^\infty(X);\,P(\partial)f=0\}\) has the linear topological invariant \((\Omega)\) of Vogt and Wagner which plays a prominent role when dealing with the question of surjectivity of \(P(\partial)\) on spaces of vector valued smooth functions. Via Grothendieck-Köthe duality this can be interpreted as an abstract version of another classical theorem from complex analysis, namely Hadamard's Three Circles Theorem. See here for a PDF version of this abstract.
In this talk we study the notion of input-to-state stability for linear systems on Banach spaces with a possibly unbounded control operator. This class of systems includes for instance boundary control problems, which are described by evolution partial differential equations. Our main interest lies in the connection between input-to-state stability and integral input-to-state stability for bounded inputs. We show that the latter is equivalent to input-to-state stability with respect to some Orlicz space. For the strong versions of those stability notions this equivalence in general does not hold. Assuming that the semigroup associated with the system is strongly stable, we show that the infinite-time admissibility with respect to an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable. The converse fails in general. This talk is based on joint work with Birgit Jacob, Jonathan R. Partington, and Felix L. Schwenninger.
TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund
Sie finden uns auf dem sechsten Stock des Mathetowers.
Janine Textor (Raum M 620)
Tel.: (0231) 755-3063
Fax: (0231) 755-5219
Mail: janine.textor@tu-dortmund.de
Bürozeiten:
Di. und Do. von 8 bis 12 Uhr
Home Office:
Mo. und Fr. von 8 bis 12 Uhr