In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X, \mathsf{d}, \mathfrak{m})$, $1 \lt q \lt \infty$, in metric measure spaces $(X, \mathsf{d}, \mathfrak{m})$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $\Gamma$-convergence; this result extends Cheeger's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $\mathfrak{m}$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

In these lectures, we review a series of optimal transport (OT) problems of growing complexity. Surprisingly enough, in this seemingly narrow framework, we will encounter nonlinear PDEs of very different type, such as the Monge–Ampère équation, the Euler equations of incompressible fluids, the hydrostatic Boussinesq equations in convection theory, the Born–Infeld equation of electromagnetism, showing the hidden richness of the concept of optimal transportion

We prove short-time existence of $\varphi$-regular solutions to the planar anisotropic curvature flow, including the crystalline case, with an additional forcing term possibly unbounded and discontinuous in time, such as for instance a white noise. We also prove uniqueness of such solutions when the anisotropy is smooth and elliptic. The main tools are the use of an implicit variational scheme in order to define the evolution, and the approximation with flows corresponding to regular anisotropies.

Existence of solutions to systems of parabolic equations obtained from polyconvex functions, remains a challenge in PDEs. In the current notes, we keep our focus on a variational problem which originates from a discretization of such a system. We state a duality result for a functional whose integrand is polyconvex and fails to satisfy growth conditions imposed in the standard theory of the calculus of variations.

The current notes are based on a work with Roméo Awi [3] and on a lecture we gave at the meeting “Variatonal Methods for Evolving Objects”, July 30–August 03, 2012, Sapporo, Japan. We express our gratitude to the organizers of the meeting for their support and generous invitation.

Many reasons can be cited for the desire to harness the power of nonlinear anisotropic diffusion in image processing. Perona and Malik proposed one of the pioneering models which, while numerically viable, proves mathematically ill-posed. This discrepancy between its analytical properties and those of its numerical implementations spurred a significant amount of research in the past twenty years or so. An overview of the latter is the topic of this article.

We survey recent results that characterize the dynamics, in a certain asymptotic limit, of interfaces in certain semilinear hyperbolic equations, as well as vortex filaments in semilinear hyperbolic systems. This survey includes a lengthy discussion of heuristic considerations, together with some complete proofs in simple model cases. We also present some novel recent approaches to problem that geometric evolution problem of timelike extremal submanifolds of Minkowski space, which governs the asymptotic dynamics of interfaces and vortex filaments.

We study degenerate and singular parabolic equations in one space dimension. The emphasis is put on the regularity of solutions and the creation as well as the evolution of facets. Facets are understood as flat parts of the graph of solutions being a result of extremely high singularity. The systems, which we consider, arise from the theory of crystals.

We present the foundations of the theory of functions of bounded variation and sets of finite perimeter in abstract Wiener spaces.