Professor: Hugo Tavares, office 5.44, Pavilhão de Matemática, IST

Email: hugo.n.tavares@tecnico.ulisboa.pt

Schedule:

Reading course. Sessions will be scheduled on a weekly basis.

Assesment methods: 

• 4 written homeworks (60% of the final grade). Weeks #4 (Sep 29 - Oct 3), #7 (Oct 20-24), #10 (24-28 Nov), #13 (Dec 15-19) 

• One final project with written work and presentation during the examination period (40% of the final grade)

Exercise sheets: 

• Exercise sheet 1

• Exercise sheet 2

• Exercise sheet 3

• Exercise sheet 4

Homeworks:

• Homework 1 (due October 17, at 20h)

• Homework 2 (due November 8, at 20h)

Slides of the 1st lecture: original version, and a version without breaks

Extra: In which contexts does the Nonlinear Schrödinger Equation (NLS) appear?

• A deduction of the NLS motivated by nonlinear optics: check pages 3-5 of this master thesis. Check also this wikipedia page on nonlinear optics. 

• For a motivation of the NLS in Bose-Einstein condensation, check out this paper. See also this wikipedia page on the Gross-Pitaevskii equation.

Extra: some history about Sobolev spaces and PDEs: 

• article Partial Differential Equations in the 20th century (see in particular section 4 on the Dirichlet's principle)

• article Remarks on the pre-history of Sobolev spaces (see in particular section 1 on the Dirichlet's principle)

Pre-Requisites: Functional Analysis and Partial Differential Equations (Master level)

Context and objectives

The Partial Differential Equations (PDEs) course in the master's degree focuses on linear equations, both because they are important and because this curricular unit is a first exposure to these type of problems. However, many of the current open problems in PDEs, at the forefront of research, are nonlinear. The goal of the Topics in DE&DS course plan is to provide students with an overview of the techniques used for nonlinear stationary problems (elliptic equations) and nonlinear evolution problems (dispersive equations). The focus will be mainly on nonlinear Schrödinger equations, where the two theories are interconnected and powerful results can be achieved.

The ideas and techniques presented in the course provide an important perspective on general PDEs that may be useful later. We will also illustrate its connection with other areas, such as Calculus of Variations, Functional Analysis and Geometry. For students who wish to explore these topics in their thesis (Masters or PhD), several problems can be provided. The course is accessible to any student who has attended the PDE course in their master's degree and knows some basic Functional Analysis.

Detailed Program

Nonlinear stationary (elliptic) equations

1. Functional Analysis: quick review of known facts and presentation of new results

• Lebesgue spaces in euclidean domains. Hölder and Interpolation inequalities. Dominated convergence, Fatou's lemma. The space L^2 of functions with complex values as a real Banach space.

• Sobolev Spaces. Poincaré's, Sobolev and Gagliardo-Nirenberg inequalities. Weak convergence, Banach-Alaoglu theorem and applications in the context of Lebesgue and Sobolev spaces. Rellich-Kondrachov compactness result for Sobolev embeddings.

2. Elliptic Equations: essential toolkit

• Maximum principles for problems with general second order elliptic operators. Hopf's lemma. Maximum principles via energy methods for divergence-type operators.

• Regularity theory: how to go from weak to classical solutions. Statements of the Schauder and L^p regularity theory. The bootstrap method. Brezis-Kato interpolation technique.

• Decay at infinity of solutions of certain classes of problems defined in the whole space.

3. Variational methods for semilinear elliptic problems. Critical point theory.

• Weak solutions as critical points of a functional. Differential calculus for functionals. Constrained minimizers and the Lagrange Multiplier Theorem on Hilbert manifolds.

• Variational Characterizations of the eigenvalues of the Laplace operator (Courant-Fisher-Weyl min-max principle).

• Problems with compactness (bounded domains, Sobolev subcritical problems): global minimization; constrained minimization and Nehari manifold methods to solve semilinear elliptic problems. Min-max methods: deformation lemma and the Mountain Pass Theorem. Ground-states.

• Problems without compactness (in the whole space and critical exponents). P.L. Lions' Concentration Compactness principle. Best constants in Sobolev and Gagliardo-Nirenberg's inequalities (and associated elliptic PDEs)

Nonlinear evolution (dispersive) equations

4. More tools from functional analysis

• Bochner's integral. 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.

• Schwarz function space. Distributions and tempered distributions. The Fourier transform: definition, properties, Parseval's identity. Extension to L^2 and definition of Fourier transform of a tempered distribution.

• Aplications to the resolution of linear PDE's with constant coefficients (in particular, to the linear Schrödinger equation without potentials). Nonhomogeneous problems and Duhamel's formula. Strichartz estimates.

5. The nonlinear Schrödinger equation (NLS)

• Invariances and conservation laws. Considerations regarding local well-posedness. Criticality. 

• Local well-posedness (existence, uniqueness and continuous dependence) in algebra spaces. Blowup alternative.

• Local well-posedness in H^1. Global existence in defocusing or L^2-subcritical cases. Global existence for small data in the L^2-critical case. Relation with Gagliardo-Nirenberg's inequality. Virial's identity and blow-up phenomena.

• Ground-states revisited in the context of the NLS: variational characterizations, instability in the L^2-critical and supercritical cases, orbital stability in the L^2-subcritical case.

Main References

Nonlinear stationary (elliptic) equations

• M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners - Existence Results via the Variational Approach, Springer-Verlag London, 2011.

• Q. Han, F. Lin, Elliptic Partial Differential Equations. Courant Lecture Notes 1, American Mathematical Society, 2000.

• M. Willem; Minimax Theorems, Birkhauser Verlag, 1996.

Nonlinear evolution (dispersive) equations

• T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, American Mathematical Society, Providence, RI, 2003.

• F. Linares, G. Ponce. Introduction to Nonlinear Dispersive Equations, Springer-Verlag, 2015.

Secondary References

Nonlinear stationary (elliptic) equations

• A. Ambrosetti, A. Malchiodi, Nonlinear analysis and semilinear elliptic problems. Cambridge Studies in Advanced Mathematics 104. Cambridge University Press, 2007.

• D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer- Verlag, Berlin (Reprint of the 1998 edition), 2001.

Nonlinear evolution (dispersive) equations

• R. Danchin, P. Raphaël, Solitons, explosion and dispersion - une introduction à l’étude des ondes non linéaires, École Polytechnique, 2015.

• T. Tao, Nonlinear dispersive equations: local and global analysis. American Mathematical Society, 2005.

• A. Haraux, T. Cazenave, An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications. 1998.