Quasiconvexity and weak convergence in nonlinear analysis

The present thesis addresses a broad range of weak convergence problems arising in Nonlinear Analysis. The thesis is divided into three related but essentially independent parts.
In the first part we study the general theory of Compensated Compactness. We begin by giving a new characterization of partial differential operators with constant rank, a mild non-degeneracy assumption which plays an important role in the theory. We then characterize completely the class of nonlinearities which are weakly continuous with respect to constant rank PDEs; in particular, we prove that it agrees with the class of nonlinear operators with Hardy space integrability, answering positively a question by Coifman–Lions–Meyer–Semmes. As an application of this theory we study homogenization problems, both with and without constant rank assumptions. In the constant rank setting we revisit the classical G-closure problem and discuss its connection with quasiconvexity. In the non-constant rank setting we study a homogenization problem for the Einstein vacuum equations in General Relativity: under some symmetry and gauge assumptions, we prove a conjecture by Burnett from 1989 which describes the effective behaviour of a sequence of vacuum space-times. This part of the thesis contains joint work with Jan Kristensen (University of Oxford), Bogdan Raită (MPI Leipzig), Matthew Schrecker (UCL) and Rita Teixeira da Costa (University of Cambridge).
The second part of this thesis is concerned with quasiconvexity in the classical, curl-free, Calculus of Variations. We contribute to the understanding of the geometry of the class of quasiconvex functions, in particular through its extremal points. We prove a Choquet-type theorem for quasiconvex functions and we provide several examples of extremal quasiconvex functions, proving in particular a conjecture made by Šverák. We then further investigate, through numerical experiments, the relationship between rank-one convexity and quasiconvexity, particularly in low dimensions. We also give a concise proof of Ornstein’s L1 non-inequality in low dimensions. This part of the thesis contains joint work with Daniel Faraco (Universidad Autónoma de Madrid) and Rita Teixeira da Costa (University of Cambridge).
The third part of this thesis deals with low regularity problems for nonlinear underdetermined PDEs. We mostly focus on the prescribed Jacobian equation, although applications to energy-dissipative so- lutions of the incompressible Euler equations are also discussed. Concerning the prescribed Jacobian equation, we prove an ill-posedness result for the Dirichlet problem. We also study the uniqueness and symmetry properties of energy minimisers with prescribed Jacobian, concluding that in general they are non-unique and non-symmetric. These results answer several questions posed by Hélein, Hogan– Li–McIntosh–Zhang and Ye in the 1990s and provide some of the first results concerning low regularity solutions of the Jacobian equation. We also prove a nonlinear version of the classical Open Mapping Theorem from Functional Analysis. Our result applies to a wide range of PDEs and, in particular, it applies to the weakly continuous nonlinearities characterized in the first part of this thesis, of which the Jacobian determinant is a particular example. As consequences of this nonlinear Open Mapping Theo- rem, we prove: i) a partial selection criterion for solutions of the Jacobian equation in the critical Sobolev space, and ii) generic non-existence of weak solutions to the incompressible Euler equations over Rn with fastly decaying kinetic energy. This part of the thesis contains joint work with Lukas Koch (University of Oxford) and Sauli Lindberg (Aalto University).

Keywords: Homogenization
Compensated Compactness
Jacobian
Quasiconvexity
Weak convergence
Advisor: Kristensen, J