### You can download some lecture notes and contributed talks in pdf format.

# Lectures

The lectures and conferences will take palce in FORUM G the first week and in the Solvay amphitheater during the second week. The contributed talks are scheduled in FORUM G, E or H during the first week and in the Solvay amphitheater or rooms NO707 or NO708 during the second week. The workshop of Saturday June 2 is scheduled in FORUM G.The complete timetable is available here in PDF format. You will receive a printed copy the first day of the meeting.

## 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

# Conferences

- Antoine Gloria, Strong ellipticity of nonlinear elastic materials and discrete homogenization - June 2
- Marco Squassina, Some results on the quasi-linear Schrodinger equation - June 2
- Charles Stuart, Asymptotic linearity and Hadamard differentiability - June 2
- Michel Willem, Symmetry of ground states of coupled Schrodinger equations - June 2
- Enrico Serra, On a conjecture of De Giorgi concerning nonlinear wave equations - June 5
- Jean Mawhin, Radial solutions of Neumann problems for periodic perturbations of the extrinsic mean curvature operator on an annulus - June 5

# Communications

The contributed talks will be scheduled on May 30, May 31 and June 4.

Session 1A - May 30, 4:00 pm : Carlo Mercuri - Sergey Kolonitskii - Alessio Pomponio - Benedetta Noris - Maria-Magdalena Boureanu

Session 1B - May 30, 4:00 pm : Alberto Saldana - Filomena Feo - Barbara Brandolini - Anna Zhigun

Session 2A - May 31, 4:30 pm : Isabel Coelho - Chiara Corsato - Sabrina Rivetti - Andrea Sfecci

Session 2B - May 31, 4:30 pm : Sara Barile - Christopher Grumiau - Tiago Picon - Naoki Shioji

Session 3A - June 4, 11:00 am : Giovany Figueiredo - Liliane Maia - Marcelo Furtado - Elvise Berchio

Session 3B - June 4, 11:00 am : Abbasali Mohammadi - Joseph Shomberg - Jesus Montejo Gamez

Session 4A - June 4, 4:00 pm : Tingjian Luo - Gaetano Siciliano - Pietro d'Avenia - Gilles Evequoz - Michael Melgaard

Session 4B - June 4, 4:00 pm : Damian Wisniewski - Alexander Nazarov - Manon Nys - Bertrand Desmons

### Titles of the contributions

- Marcos Pimenta, Multiplicity of solutions for a biharmonic equation with subcritical or critical growth
- Giovany Figueiredo, On the number of solutions of NLS equations with magnetics elds in expanding domains
- Christopher Grumiau, Lane Emden problems: asymptotic behavior of low energy nodal solutions
- Anna Zhigun, On a 'balance' condition for a class of PDEs including porous medium and chemotaxis effect
- Liliane Maia, Weakly coupled nonlinear Schrodinger systems: the saturation effect
- Alberto Saldana, Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains
- Peng Feng, Boundary blow up solutions to elliptic equations and systems
- Tingjian Luo, Existence and instability of standing waves with prescribed norm for a class of Schrodinger-Poisson equations
- Filomena Feo, Uniqueness result for nonlinear parabolic equations with lower order terms
- Joseph Shomberg, On damped semilinear wave equations with singularly perturbed boundary conditions
- Naoki Shioji, Partial radial symmetry of positive solutions for semilinear elliptic equations in a disc and its application to the H\'enon equation
- Damian Wisniewski, Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains
- Barbara Brandolini, Local behaviour of singular solutions for nonlinear elliptic equations in divergence form
- Elvise Berchio, On the solutions set to some Robin problems
- Alexander Nazarov, On the solvability of a BVP generated by the Maz'ya-Sobolev inequality
- Abbasali Mohammadi, A minimization problem related to the principal eigenvalue of the s-wave Schrodinger operator
- Gaetano Siciliano, schrodinger-poisson system with non constant interaction
- Marcelo Furtado, Multiple solutions for nonlinear elliptic equations with fast increasing weight and critical growth
- Benedetta Noris, Multiplicity of solutions of a p-Laplacian equation via the sub-supersolutions method
- Sergey Kolonitskii, Multiplicity of solutions to the Dirichlet problem for a supercritical equation with p-laplacian
- Maria-Magdalena Boureanu, Existence and multiplicity results for Neumann problems with variable exponents
- Tiago Picon, L^{1} estimates for complex elliptic systems of vector fields
- Gilles Evequoz, Entire solutions to nonlinear scalar field equations with indefinite linear part
- Pietro d'Avenia, Generalized Schrodinger-Poisson type systems
- Sara Barile, Existence and Multiplicity Results for some Elliptic Systems in Unbounded Cylinders
- Alessio Pomponio, Orlicz-Sobolev embeddings and applications to quasilinear elliptic equations in R^N
- Jesus Montejo Gamez, On the initial--boundary problem associated with a nonlinear dissipative Schroedinger equation
- Chiara Corsato, Positive solutions of the Minkowski-curvature equation, part I
- Isabel Coelho, Positive solutions of the Minkowski-curvature equation, part II
- Atul Kumar, Horizontal Solute Transport from a Pulse Type Source along Temporally and Spatially Dependent Flow: Analytical Solution
- Bertrand Desmons, Wrinkling of thin membranes laying on fluid substrates under small compression
- Michael Melgaard, Solutions to locally compact variational problems with nonlocal operators
- Sabrina Rivetti, Periodic solutions of a capillarity equation in the presence of lower and upper solutions
- Andrea Sfecci, A nonresonance condition for radial solutions of a nonlinear Neumann elliptic problem