A glimpse of vectorial Calculus of Variations in L∞ and fully nonlinear PDE systems through the non-expert's keyhole

DATES: 8-12 May 2017 (5 sessions)
TIME: 09:30 - 11:30 (a total of 10 hours)

In this course I will expound on the main concepts and methods emerging in the rapidly developing area of Calculus of Variation in the space L∞, focusing mostly on the vector-valued and the higher order case.

The central point of this theory is to study minimisation problems for functionals modelled after the sup-norm of the derivatives of admissible functions. This area is motivated by important problems arising elsewhere in pure and applied mathematics (e.g., optimal Lipschitz extensions, quasi-conformals maps, Riemannian geometry, data assimilation, PDE-constrained optimisation-). A key feature is that the analogues of the "Euler-Lagrange" equations which describe critical points in L∞ are nondivergence PDEs and actually fully nonlinear and higher order with discontinuous coefficients, rendering the use of all standard PDE approaches inapplicable in defining and studying generalised solutions to the equilibrium equations.

The scalar case was pioneered by G.Aronsson in the 1960s and nowadays it has evolved to a fairly complete theory (see the 2015 Springer Brief - An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞" of the lecturer corresponding to a synonymous 2013 BCAM course devoted to the scalar case). The vectorial and higher order case was initiated some 50 years later in the early 2010s by the lecturer and is developing rapidly due to both the theoretical importance as well as the relevance to real-world applications.

PROGRAMME
- Session 1
Gentle introduction to conventional Calculus of Variations, historical remarks and motivation of the L-infinity theory (optimal Lipschitz extensions, Quasi-conformals maps, Data Assimilation, PDE constrained optimisation, Riemannian Geometry).
- Session 2
The main objects-concepts in the vectorial case and connections to the scalar case (supremal 1st order functionals in the vectorial case, the infinity-Laplace system and the general PDE system, absolute minimisers, basic properties of classical solutions).
- Session 3
D-solutions and results on the vectorial case (introduction to Young measures and L^p spaces over Banach spaces, generalised solutions, the Baire Category method / Convex Integration, existence and (non-)uniqueness for the Dirichlet problem, etc).
- Session 4
The main objects and concepts in the higher order case (supremal 2nd order functionals, the infinity-Polylaplacian/BIlaplacian and the general PDE, absolute minimisers, basic properties of classical solutions).
- Session 5
D-solutions and results on the higher order case (generalised solutions, the Baire Category method / Convex Integration, existence-uniqueness for the Dirichlet problem, etc.).

PREREQUISITES
- Essential: Basic undergraduate Analysis.
- Desirable: Knowledge of Measure theory, Sobolev spaces, conventional Calculus of Variations and elliptic PDEs is desirable but by no means necessary.

*Registration is free, but inscription is required before 3rd May: So as to inscribe send an e-mail to roldan@bcamath.org. Student grants are available. Please, let us know if you need support for travel and accommodation expenses.

• BCAM Course KATZOURAKIS