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# Matematiska kollokviet

Organiserat av Anders Björn, Milagros Izquierdo, Vladimir Kozlov och Hans Lundmark.

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## Onsdag 23 augusti 2017, Petros Petrosyan, Yerevan State University, Armenien

Talare: Petros Petrosyan, Yerevan State University, Armenien

Titel: Number of palettes in edge-colorings of graphs

Tid och plats: Onsdag 23 augusti 2017, Hopningspunkten, 13.15–14.15

Sammanfattning:

A proper edge-coloring of a graph $G$ is a mapping $\alpha: E(G)\rightarrow \mathbb{N}$ such that $\alpha(e)\neq \alpha(e^{\prime})$ for every pair of adjacent edges $e,e^{\prime}\in E(G)$. If $\alpha$ is a proper edge-coloring of a graph $G$ and $v\in V(G)$, then the palette of a vertex $v$, denoted by $S\left(v,\alpha \right)$, is the set of all colors appearing on edges incident to $v$. For a proper edge-coloring $\alpha$ of a graph $G$, we define $S(G,\alpha)$ as follows: $S(G,\alpha)=\{S\left(v,\alpha \right)\colon\,v\in V(G)\}$. For every graph $G$ and  its proper edge-coloring $\alpha$, we have $1\leq \vert S(G,\alpha)\vert\leq \vert V(G)\vert$. In 1997, Burris and Schelp introduced the concept of vertex-distinguishing proper edge-colorings of graphs. A proper edge-coloring $\alpha$ of a graph $G$ is a vertex-distinguishing edge-coloring if for every pair of distinct vertices $u$ and $v$ of $G$, $S\left(u,\alpha \right)\neq S\left(v,\alpha \right)$. This means that if $\alpha$ is a vertex-distinguishing edge-coloring of $G$, then $\vert S(G,\alpha)\vert=\vert V(G)\vert$. On the other hand, recently Hor\vn\'ak, Kalinowski, Meszka and Wo\'zniak initiated the study of the problem of finding proper edge-colorings of graphs with the minimum number of distinct palettes. For a graph $G$, they define the palette index $\check{s}(G)$ of a graph $G$ as follows: $\check{s}(G)=\min_{\alpha}\vert S(G,\alpha)\vert$, where minimum is taken over all possible proper edge-colorings of $G$. In this talk we will give a survey of the topic and present a recent progress in the study of palette indices of graphs.

## Onsdag 30 augusti 2017, Jari Taskinen, Helsingfors Universitet, Finland

Talare: Jari Taskinen, Helsingfors Universitet, Finland

Titel: Band-gap spectra of some elliptic equations and systems on  waveguides

Tid och plats: Onsdag 30 augusti 2017, Hopningspunkten, 13.15–14.15

Sammanfattning:  We consider the band-gap structure of the essential spectrum of some elliptic spectral problems on  periodic 2- and 3-dimensional waveguides. In the recent paper with S. Nazarov [1] we study the linearized piezoelectricity system on waveguides with thin structures, which are created by thin ligaments connecting (infinitely many, translated copies of) bounded cells. We establish the existence of an arbitrary  number of gaps, if the connecting ligaments of the cells are thin enough. The problem is non-selfadjoint, thus we apply a self-adjoint reduction scheme; also the mere existence of the  band-gap structure for the essential spectrum needs a new proof, which we able to provide.

In the work [2] with F. Bakharev we study the linearized elasticity system for waveguides, the geometry of which is similar to the above situation. We perform an asymptotic
analysis to obtain quite precise information on the position of the spectral bands.

Finally, in the project [3] we study the Laplace-Dirichlet problem in the plane which is perforated by a periodic lattice of  discs with radius $r > 0$. Applying  Floquet-Bloch-Gelfand-techniques we show that the FBG-eigenvalues depend real analytically on the geometric parameter $r$. This leads to a non-existence result for eigenvalues of infinite multiplicity.

[1] S. Nazarov, JT: Spectral gaps for periodic piezoelectric waveguides, Z. Angew. Math. Phys. 66,  6 (2015),  3017-3047.

[2] F. Bakharev, JT: Bands in the spectrum of a periodic elastic waveguide. To appear in  Z. Angew. Math. Phys.

[3] M. Lanza de Cristoforis, P. Musolino, JT: work in preparation.

## Onsdag 6 september 2017, Ugo Gianazza, University of Pavia, Italien

Talare: Ugo Gianazza, University of Pavia, Italien

Titel:  A self-improving property of degenerate parabolic equations of porous medium-type

Tid och plats: Onsdag 6 september 2017, Hopningspunkten, 13.15–14.15

Sammanfattning: We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse Hölder inequality in  suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calderón-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution u.

This is a joint work with Sebastian Schwarzacher of Charles University.

Sidansvarig: milagros.izquierdo@liu.se
Senast uppdaterad: Tue Aug 01 12:39:00 CEST 2017