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Nicolas Vauchelet (Universidad Paris 6)

Mathematical study of a cell model for tumor growth : travelling front and incompressible limit

We consider mathematical models at macroscopic scale to describe tumor growth. In this view, tumor cells are considered as an elastic material subjected to mechanical pressure. Two main classes of model can be encountered: those describing the dynamics of tumor cells density and those describing the dynamic of the tumor thanks to the motion of its domain. These latter models are free boundary problem. We will show that such free boundary problem of Hele-Shaw type can be derived thanks to an incompressible limit from models describing the dynamics of cells density. Moreover, for this model we study the existence of travelling waves, allowing to describe the spread of the tumor.


Pierpaolo Esposito (Universidad de Roma Tre)

Equilibria of point-vortices on closed surfaces

I will discuss the existence of equilibrium configurations for the Hamiltonian point-vortex model on a closed surface. Its topological properties determine the occurrence of three distinct situations, corresponding topologically to the sphere, to the real projective plane and to the remaining cases. As a by-product, new existence results are obtained for the singular mean-field equation with exponential nonlinearity.

Joint work with T. D’Aprile.


Oscar Agudelo (University of West Bohemia)

Singularly perturbed Allen-Cahn equation with catenoidal nodal sets

In this lecture we review some recent results concerning existence and asymptotic behavior of solutions to the singularly perturbed problem

alpha^2 Delta u + u(1-u^2)=0, in Omega

where Omega subset mathbb{R}^N is either a smooth bounded domain or the entire space and Ngeq 2. We take advantage of the deep connection between the equation above and the theory of minimal surfaces to study asymptotic profiles of the solutions. Particular attention is paid to solutions with catenoidal nodal set.