We investigate a mixed problem with Robin boundary conditions for a diffusion-reaction equation. We investigate the problem in the sublinear, linear and super linear cases, depending on the nonlinear part. We obtain relations between the parameters of the problem which are sufficient conditions for the existence of generalized solutions to the problem and, in a special case, for their uniqueness. The proof relies on a general existence theorem by Soltanov. Finally we investıgate the time-behaviour of solutions. We show that boundedness of solutions holds under some additional conditions as t is convergent to infinity. This study is joint work with Kamal Soltanov (Hacettepe University).
Nematic elastomers are rubbery elastic solids made of cross-linked polymeric chains with embedded nematic mesogens. Their mechanical behaviour results from the interaction of electro-optical effects typical of nematic liquid crystals with the elasticity of a rubbery matrix. We show that the geometrically linear counterpart of some compressible models for these materials can be justified via Gamma-convergence. A similar analysis on other compressible models leads to the question whether linearised elasticity can be derived from finite elasticity via Gamma-convergence under weak conditions of growth (from below) of the energy density. We answer to this question for the case of single well energy densities.
We discuss Ogden-type extensions of the energy density currently used to model nematic elastomers, which provide a suitable framework to study the stiffening response at high imposed stretches.
Finally, we present some results concerning the attainment of minimal energy for both the geometrically linear and the nonlinear model.
I will be concerned with initial-boundary value problems for systems of conservation laws in one space variable. First, I will go over some of the most relevant features of these problems. In particular, I will stress that different viscous approximation lead, in general, to different limits.
Next, I will discuss possible ways of characterizing the limit of a given viscous approximation. Also, I will establish a uniqueness criterion that allows to conclude that the limit of a self-similar approximation introduced by Dafermos et al. actually coincide with the limit of the physical viscous approximation. Finally, if time allows I will mention consequences on the design of numerical schemes. The talk will be based on joint works with S. Bianchini, C. Christoforou and S. Mishra.
Crystalline solids are descibed by a material manifold endowed
with a certain structure which we call crystalline. This is characterized by
a canonical 1-form, the integral of which on a closed curve in the material manifold
represents, in the continuum limit, the sum of the Burgers vectors of all the dislocation lines
enclosed by the curve. In the case that the dislocation distribution is uniform, the material manifold
becomes a Lie group upon the choice of an identity element. In this talk crystalline solids
with uniform distributions of the two elementary kinds of dislocations, edge and screw dislocations,
shall be considered. These correspond to the two simplest non-Abelian Lie groups, the affine group
and the Heisenberg group respectively. The statics of a crystalline solid are described in terms of a
mapping from the material domain into Euclidean space. The equilibrium configurations correspond
to mappings which minimize a certain energy integral. The static problem is solved in the case of
a small density of dislocations.
Several shape optimization problems, e.g. in image processing, biology, or discrete geometry, involve the Willmore functional, which is for a surface the integrated squared mean curvature. Due to its singularity, minimizing this functional under constraints is a delicate issue. More precisely, it is difficult to characterize precisely the structure of the minimizers and to provide an explicit
formulation of their energy. In a joint work with Giacomo Nardi (Paris-Dauphine), we have studied an "integrated" version of the Willmore functional, i.e. a version defined for functions and not only for sets. In this talk, I will describe the tools, based on Young measures and varifolds, that we have introduced to address the relaxation issue. I will also discuss some connections with the phase-field numerical approximation of the Willmore flow, that we have investigated with Elie Bretin (Lyon) and Edouard Oudet (Grenoble).
We review potential theoretic aspects of degenerate parabolic PDEs of p-Laplacian type.
Solutions form a similar basis for a nonlinear parabolic potential theory as the solutions of the heat
equation do in the classical theory. In the parabolic potential theory, the so-called superparabolic
functions are essential. For the ordinary heat equation we have supercaloric functions. They are defined
as lower semicontinuous functions obeying the comparison principle. The superparabolic
functions are of actual interest also because they are viscosity supersolutions of the equation. We discuss
their existence, structural, convergence and Sobolev space properties. We also consider the
definition and properties of the nonlinear parabolic capacity and show that the infinity set of a superparabolic
function is of zero capacity.