Gibson 1st Floor SR

An overview is given of some key issues and definitions in the Calculus of Variations, with a focus on lower semicontinuity and quasiconvexity. Some well known results and instructive counterexamples are also discussed. We then move to consider variational problems in the BV setting, and present a new lower semicontinuity result for quasiconvex integrals of subquadratic growth. The proof of this requires some interesting techniques, such as obtaining boundedness properties for an extension operator, and exploiting fine properties of Sobolev maps.

We describe several bifurcation properties corresponding to various classes of nonlinear elliptic equations The purpose of this talk is two-fold. First, it points out different competition effects between the terms involved in the equations. Second, it provides several non standard phenomena that occur according to the structure of the differential operator.

This is a joint work with Craig Evans. We study the partial regularity of minimizers for certain functionals in the calculus of variations, namely the modified Landau-de Gennes energy functional in nematic liquid crystal theory introduced by Ball and Majumdar.

We prove existence of a global semigroup of conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. The equation was derived by Saxton as a model for liquid crystals. This equation shares many of the peculiarities of the Hunter–Saxton and the Camassa–Holm equations. In particular, the equation possesses two distinct classes of solutions denoted conservative and dissipative. In order to solve the Cauchy problem uniquely it is necessary to augment the equation properly. In this talk we describe how this is done for conservative solutions. The talk is based on joint work with X. Raynaud.

We will give a survey on some recent results on travel tomography which consists in determining the index of refraction of a medium by measuring the travel times of sound waves going through the medium. In differential geometry this is known as the boundary rigidity problem. We will also consider the related problem of tensor tomography which consists in determining a function, a vector field or tensors of higher rank from their integrals along geodesics.