Variational methods for PDEs and nonlocal problems


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Aim: This course aims to give an introduction from the basics to modern applications of variational methods, particularly for partial differential equations and discrete nonlocal minimisation problems in the physical sciences. We  focus on problems from continuum mechanics, including recent developments for convex integration, but also on nonlocal energies for optimal point or crystal configurations.

Contents: We cover some essential basic and advanced topics in calculus of variations and nonlinear PDEs, as well as their applications. After the course the student should know key ideas in a broad range of topics, as they are relevant in their research or in related pure/numerical analysis research seminars. In particular, we expect to cover the following topics:

  1. A user’s guide to Sobolev spaces.
  2. Aspects of the calculus of variations: lower semicontinuity, convexity.
  3. Applications: equations of continuum mechanics, martensitic phase transitions.
  4. Numerical discretisation: Finite element approximation of nonlinear PDEs.
  5. Convex integration and non-uniqueness of weak solutions.
  6. Nonlocal and discrete minimization problems: Log-gases and optimal point configurations.

Some references:

  • S. Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer, 2015.
  • C. De Lellis, L. Szekelyhidi Jr., John Nash's nonlinear iteration, Memorial Volume for John Nash, 2017.
  • M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, 2008.
  • E. B. Saff, V. Totik, Logarithmic potentials with external fields, Springer, 2013.

Further references for special topics will be provided in the lectures.

Student talks: Interested students will give a 60-minute lecture on a topic of their choice, ideally a topic related to their research interests.