Variational approach to nonlinear elliptic problems

We consider the existence of least energy solutions of nonlinear elliptic problems via variational approach. We deal with both scalar problems and systems of elliptic equations. We also give applications to singularly perturbed Schrodinger equations. More precisely, in first half of my lecture, we will mainly deal with scalar field equations and give another proof of the fundamental result of Berestycki and Lions using Mountain Pass theorem (without introducing constraint problems). We will also introduce some technique from the theory of dynamical systems to deal with N=1 (especially for systems). If time allows, we will also deal with singular perturbation problems.

Lectures on May 30,31 and June 1

Arithmetic progressions and near equality in affine invariant inequalities

These lectures will survey progress in understanding those functions which extremize, and especially those which nearly extremize, certain inequalities. Among these are Young's convolution inequality, the Riesz-Sobolev rearrangement inequality, the Brunn-Minkowski inequality, and an inequality for the Radon transform. The group of all invertible affine transformations of Euclidean space is a symmetry group of each of these inequalities. Arithmetic progressions and the geometry of Euclidean space are at the heart of the analysis.

Lectures on May 30, 31 and June 1

Self-dual variational calculus: From existence and uniqueness to homogenization and control

We describe how the theories of self-dual Lagrangians and anti-symmetric Hamiltonians can be applied to give variational proofs for the existence and uniqueness of solutions to various PDEs and evolution equations. It is also applied to the study of inverse problems, optimal control, and the homogenization of equations involving monotone vector fields. It also leads to a self-dual version of Brenier's polar decomposition for general vector fields.

Lectures on May 30, 31 and June 1

The Mathematics of Liquid Crystals

Liquid crystals represent a vast and diverse class of materials which are intermediate between isotropic liquids and crystalline solids. The lecture will describe these fascinating materials and what mathematics, in particular the calculus of variations and partial differential equations, can say about them. The Landau ? de Gennes theory, in which the orientational order of the molecules is represented by a matrix-valued order parameter, will be emphasised, together with its relation to the simpler Oseen-Frank theory.

Lectures on May 31 and June 1

TBA

Lectures on June 4,5,6

Equivariant Bifurcation in Geometric Variational Problems

The first lecture will be about some abstract results on the equivariant implicit function theorem and equivariant bifurcation. In the second lecture I will discuss some applications to the case of minimal and CMC embeddings in Riemannian geometry. The third lecture will be about applications to the Yamabe problem and its generalizations.

Lectures on June 4,5,6

Stability of solitons and multi-solitons via the variational method. Application to the Gross-Pitaevskii equation.

Stability of solitary waves has long been an intensive area of research. Modern methods include a) inverse scattering transforms and/or Riemann-Hilbert problems, when the problem at hand is integrable, and b) the more robust variationnal method, in other cases. In these lectures, I will start by reviewing some presumably interesting questions related to solitary waves that arise naturally in a number of Hamiltonian equations, like the (generalized) Korteweg - de Vries (gKdV) equation, the Non Linear Schrodinger equations (NLS), and the Toda lattice equations. Even though integrable methods can be quite powerful (e.g. notably for the integrable KdV), in some cases, even for integrable equations, the variationnal method has revealed itself as the only one yet to provide tractable qualitative information. I will next focus on the Gross-Pitaevskii (GP) equation in 1D, a defocusing NLS equation with a non vanishing condition at infinity. Equation (GP) is known to be integrable since the work of Zakharov and Shabat in 1973. Yet the variationnal method seems the method of choice. I will mainly discuss three issues for (GP) : 1) orbital stability of solitary waves: on this occasion I will refer to the powerful and well-known methods of Benjamin/Cazenave-Lions/Grillakis-Shatah-Strauss. 2) orbital stability of multi-solitons: this concerns the "orbital" stability of a train of well-separated solitons. Stability does not necessarily holds in that situation, and the orbital stability (purely variationnal) of single solitons needs to be combined with dynamical information (here a monotonicity formula). 3) asymptotic stability of solitary waves: this is a much more subtle (and stronger) form of stability which requires a good understanding of the linearized flow around a solitary wave.

Lectures on June 5,6

Topics in geometric spectral theory

Geometric spectral theory is a field of mathematics at the interface of partial differential equations, geometry and functional analysis. It has a long and fascinating history, going back to the experiments of Chladni with vibrating plates, to the ground-breaking work of Rayleigh on the theory of sound and to Kac's celebrated question "Can one hear the shape of a drum?". The aim of the mini-course is to present basic notions and fundamental results of geometric spectral theory, as well as to discuss some recent developments and open problems.

Lectures on June 25 to 29