Variational problems in a nonsmooth geometric setting

In its long history, the Calculus of Variations has developed a rich bag of words, ideas and tools to deal with nonsmoothness. This thesis presents the results of three projects where such nonsmoothness appears with a distinct geometric flavour, be it in the very objects that are described or in the domains where they are defined. The first project deals with metric measure geometry in high dimension. The first part is devoted to the study of quantities which are stable with respect to convergence in concentration, a notion well suited for sequences of spaces with unbounded dimension. The main result that we obtain is the convergence of the heat flow in presence of a uniform Ricci lower bound. In order to prove that, we show – in the same assumptions – the Gamma convergence of the slope of the entropy and the Mosco convergence of the Cheeger energy. As a byproduct, we are able to define a meaningful convergence of vector fields and of Sobolev functions and to prove the stability of the solution to continuity equations and of the Laplacian eigenvalues. The second part is devoted to the study of Gromov’s pyramids and of extended metric measure spaces, two possible notions of infinite dimensional metric measure spaces. We describe a way to associate – in a possibly non-canonical fashion – to each pyramid an extended metric measure space and to each extended metric measure space a pyramid. We also discuss a natural condition under which the two operations are consistent with each other, encompassing interesting examples such as the Wiener space. The second project deals with nonsmooth Lorentzian geometry. The first part is devoted to the definition and study of a relaxed p-Cheeger energy – obtained by relaxation – and of its gradient flow – the p-heat flow. We show that this energy produces a notion of maximal subslope and prove natural calculus rules and an interesting Kuwada-type lemma, linking the energy to an entropy functional on the Wasserstein space. Finally, we prove that under some assumptions – that we believe to be technical – our maximal subslope coincides with the one defined via test plans. The second part is devoted to the definition and study of causally convex functions and of their gradient flows. We define basic convex-analytic objects and quantities – such as the slope and the subdifferential – and prove basic regularity properties for causally convex functions. Finally, we define a notion of (p, q)-EDE gradient flow via an energy-dissipation-like property and an EVI-gradient flow mimicking a differential inclusion and prove local existence, a partial uniqueness result and that – up to a suitable reparametrization – the flows coincide. The last project deals with the study of the asymptotics of a hyperelastic-type energy functional defined on vector functions on a perforated domain, in presence of nonlinear pointwise constraints in the perforation sites. We prove the Gamma convergence to a limit functional under quite general assumptions on the energy and the constraints and conclude with some numerical simulations.