Main Goals
The main Partial Differential Course (PDE) courses available in the Masters and PhD curricula focus on linear equations. Since many open problems which lie on the edge of research in PDEs are nonlinear by nature, a gap should be filled. The plan is to to provide students with an overview of the techniques used for stationary (elliptic) and evolution problems. To exemplify, the focus will mainly be on nonlinear Schrödinger equations, where the two theories are intertwined and very strong results may be achieved.
The ideas and techniques presented in the course provide an important perspective on general PDEs which can be useful later on. We will also illustrate their connection with other areas such as Calculus of Variations, Functional Analysis and Geometry. For students who desire to pursue these topics in their (Master or PhD) thesis, several problems may be given.
Pre-Requisites
Functional Analysis and Partial Differential Equations (Master level)
Detailed Program (items with * are optional)
Functional Analysis: quick review of known facts and presentation of new results.
Lebesgue spaces; Hölder and Interpolation inequalities; Dominated convergence; Fatou's lemma.
Sobolev Spaces. Poincaré's, Sobolev and Gagliardo-Nirenberg inequalities. Weak and weak-* convergence in the context of Sobolev spaces; Banach-Alaoglu theorem and applications. Rellich-Kondrachov compactness result for Sobolev embeddings.
Variational methods for semilinear elliptic problems.
Weak solutions as critical points of a functional. Differential calculus for functionals. Constrained minimizers and the Lagrange Multiplier Theorem in Hilbert manifolds. *Variational Characterizations of the eigenvalues of the Laplace operator.
Problems with compactness (bounded domains, subcritical problems): global minimization; constrained minimization and Nehari manifold methods to solve elliptic problems. Deformation lemma and the Mountain Pass Theorem.
Problems without compactness (in the whole space and critical exponents). Lions' Concentration Compactness principle. Best constants in Sobolev and Gagliardo-Nirenberg's inequalities.
Elliptic Equations: essential toolkit.
Statements of the Schauder and L^p regularity theory. The bootstrap method: how to pass from weak to classical solutions in certain classes of problems. *Brezis-Kato interpolation technique.
*Decay at infinity of solutions of certain classes of problems.
Maximum principles for problems with second order elliptic operators. Hopf's lemma.
More tools from functional analysis
Lebesgue and Sobolev spaces with values in a Banach space. The Riesz-Thorin interpolation theorem. Young's inequality for weak-L^p kernels. Hardy-Littlewood-Sobolev inequality.
The Fourier transform: definition, properties, Parseval's identity. Extension to L^2. Resolution of linear PDE's with constant coefficients. *Stationary phase method.
The nonlinear Schrödinger equation
Considerations regarding local well-posedness. Criticality. Local well-posedness in algebra spaces. Strichartz estimates. Local well-posedness in H^1 and L^2. Persistence of regularity.
Invariances and conservation laws. Global existence in defocusing or L^2-subcritical cases. *Global existence for small data in the H^1-subcritical case. Virial's identity and blow-up phenomena.
*Pseudo-conformal transformation. *Minimal mass blow-up solution. *Concentration phenomena at blow-up time. *Scattering in the weighted energy space.
Ground-states: variational characterizations, instability in the L^2-supercritical case, orbital stability in the L^2-subcritical case.
Evaluation
6 written homeworks (60% of the final grade)
One final project with written work and presentation (40% of the final grade)