Abstract.
The course is an introduction to the theory of viscosity solutions to fully nonlinear degenerate elliptic equations.
After presenting some model problems and motivating the definitions, I will illustrate the basic properties of viscosity solutions, in particular stability and the comparison principles.
In the second part I’ll discuss two qualitative properties of subsolutions: the Strong Maximum Principle, i.e., if a subsolution in an open connected set attains an interior maximum then it is constant, and the Liouville property, i.e., if a subsolution in the whole space is bounded form above then it is constant. They are standard results for classical solutions of linear elliptic PDEs, and many extensions are known. I will show how the viscosity methods allow to turn around the difficulties of non-smooth solutions, fully nonlinear equations, and their possible degeneracies
Abstract. We present some equation of nonlocal type arising in material sciences (atom dislocations in crystals) and biology (optimal foraging). Topics will include:
Abstract.
The nonlinear Schrodinger equation is a ubiquitous model in physics, with many applications in fields as diverse as Bose-Einstein condensation or non-linear optics.
In many physical situations, the underlying space (i.e. the domain on which the solutions of the equation are defined) is essentially one-dimensional
and can be modeled as a metric graph, also called a quantum graph, i.e. a locally one–dimensional domain obtained by gluing together one–dimensional,
possibly unbounded intervals, the edges, through the identification of some of their endpoints, the vertices.
The mathematical study of this type of model is very recent and is growing incredibly.
As when the non-linear Schrodinger equation is placed on the entire space the study of stationary solutions is the subject of special attention.
One can then focus either on solutions with a given frequency or on solutions having a prescribed mass, namely a prescribed L2 norm.
This second type of solution is particularly interesting from a physical point of view, as this quantity is preserved over time.
In this course, we will concentrate on this second type of solution. First, we will look at the mass sub-critical case,
whose study naturally leads us to consider minimization problems. Secondly, we will study the mass-supercritical case,
where a minimization approach is no longer possible. In particular, we will present the various tools, such as variational methods,
blow-up techniques or Morse index estimates, that are needed to deal with this case.
Abstract. We present some results about nonlocal perimeters and their applications to models of capillarity and long-range phase transitions. Topics will include:
Abstract. Sobolev mappings arise naturally in many partial differential equations and calculus of variations problems arising in physical models, geometry and computer graphics. Although they can be naturally defined as Sobolev functions satisfying almost everywhere a nonlinear manifold constraint, not all the properties of classical Sobolev spaces pass to the corresponding spaces of mappings. In particular the approximation by smooth maps and the surjectivity of the traces on fractional Sobolev spaces fail when the integrability exponent is small. The obstacles to these properties are intimately connected to obstruction theory and the finiteness or triviality of some homotopy groups of the target manifold. In these lectures, I will describe these pathologies and the nonlinear constructions leading to positive results, and relate them to the lifting and homotopy problems.