# Past Partial Differential Equations Seminar

16 June 2014
17:00
Konstantina Trivisa
Abstract
We investigate the dynamics of a class of tumor growth models known as mixed models. The key characteristic of these type of tumor growth models is that the different populations of cells are continuously present everywhere in the tumor at all times. In this work we focus on the evolution of tumor growth in the presence of proliferating, quiescent and dead cells as well as a nutrient. The system is given by a multi-phase flow model and the tumor is described as a growing continuum such that both the domain occupied by the tumor as well as its boundary evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation. Further extensions will be discussed. This is joint work with D. Donatelli.
• Partial Differential Equations Seminar
9 June 2014
17:00
Abstract
One of the holy grails of material science is a complete characterization of ground states of material energies. Some materials have periodic ground states, others have quasi-periodic states, and yet others form amorphic, random structures. Knowing this structure is essential to determine the macroscopic material properties of the material. In theory the energy contains all the information needed to determine the structure of ground states, but in practice it is extremely hard to extract this information. In this talk I will describe a model for which we recently managed to characterize the ground state in a very complete way. The energy describes the behaviour of diblock copolymers, polymers that consist of two parts that repel each other. At low temperature such polymers organize themselves in complex microstructures at microscopic scales. We concentrate on a regime in which the two parts are of strongly different sizes. In this regime we can completely characterize ground states, and even show stability of the ground state to small energy perturbations. This is work with David Bourne and Florian Theil.
• Partial Differential Equations Seminar
2 June 2014
17:00
Roger Moser
Abstract
Biharmonic maps are the solutions of a variational problem for maps between Riemannian manifolds. But since the underlying functional contains nonlinear differential operators that behave badly on the usual Sobolev spaces, it is difficult to study it with variational methods. If the target manifold has enough symmetry, however, then we can combine analytic tools with geometric observations and make some statements about existence and regularity.
• Partial Differential Equations Seminar
26 May 2014
17:00
Abstract
We present a new method to establish the rotational symmetry of solutions to overdetermined elliptic boundary value problems. We illustrate this approach through a couple of classical examples arising in potential theory, in both the exterior and the interior punctured domain. We discuss how some of the known results can be recovered and we introduce some new geometric overdetermining conditions, involving the mean curvature of the boundary and the Neumann data.
• Partial Differential Equations Seminar
19 May 2014
17:00
Abstract
In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem $$- div A(x) Du = F (x, u) \; {\rm in} \; \Omega,$$ $$u = 0 \; {\rm on} \; \partial \Omega,$$ (namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition), where the non\-linearity $F (x, u)$ is singular in $u = 0$, with a singularity of the type $$F (x, u) = {f(x) \over u^\gamma} + g(x)$$ with $\gamma > 0$ and $f$ and $g$ non negative (which implies that also $u$ is non negative). The main difficulty is to give a convenient definition of the solution of the problem, in particular when $\gamma > 1$. We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when $F(x, u)$ is non increasing en $u$. We also consider the homogenization problem where $\Omega$ is replaced by $\Omega^\varepsilon$, with $\Omega^\varepsilon$ obtained from $\Omega$ by removing many very small holes in such a way that passing to the limit when $\varepsilon$ tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" $\mu u$ appears.
• Partial Differential Equations Seminar
12 May 2014
17:00
Jean Van Schaftingen
Abstract
I will show how families of concentrating stationary vortices for the shallow water equations can be constructed and studied asymptotically. The main tool is the study of asymptotics of solutions to a family of semilinear elliptic problems. The same method also yields results for axisymmetric vortices for the Euler equation of incompressible fluids.
• Partial Differential Equations Seminar