Related papers: The Lip-lip equality is stable under blow-up
On complete metric spaces that support doubling measures, we show that the validity of a Rademacher theorem for Lipschitz functions can be characterised by Keith's "Lip-lip" condition. Roughly speaking, this means that at almost every…
We prove metric differentiation for differentiability spaces in the sense of Cheeger. As corollaries we give a new proof that the minimal generalized upper gradient coincides with the pointwise Lipschitz constant for Lipschitz functions on…
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We…
On metric spaces equipped with doubling measures, we prove that a differentiability theorem holds for Lipschitz functions if and only if the space supports nontrivial (metric) derivations in the sense of Weaver that satisfy an additional…
For each $\beta>1$ we construct a family $F_\beta$ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e.~blow-ups), and such that each element of $F_\beta$ admits a $(1,P)$-Poincar\'e inequality if and…
This paper is concerned with weak solutions of the degenerate viscous Hamilton-Jacobi equation $$\partial_t u-\Delta_p u=|\nabla u|^q,$$ with Dirichlet boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$, where $p>2$ and…
We consider the diffusive Hamilton-Jacobi equation, with homogeneous Dirichlet conditions and regular initial data. It is known from [Barles-DaLio, 2004] that the problem admits a unique, continuous, global viscosity solution, which extends…
A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with…
We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the…
Blowing up a point p in a manifold M builds a new manifold M' in which p is replaced by the projectivization of the tangent space of M at p. This well-known operation also applies to fixed points of diffeomorphisms, yielding continuous…
We give a detailed proof of some facts about the blow-up of horizontal curves in Carnot-Carath\'eodory spaces.
We establish a general inequality comparing the $F$-thresholds of a local ring and its associated graded ring. As an application, we deduce that the $F$-rationality of the graded ring descends to the local ring.
In this note, we consider an evolution coercive Hamilton-Jacobi equation posed in a domain and supplemented with a boundary condition. We are interested in proving a comparison principle in the case where the time and the (normal) gradient…
We show that length minimizing curves in Carnot-Carath\'eodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line. This is the first regularity result…
We consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For $p>2$, solutions may develop gradient…
Using the method of blow-up analysis, we obtain two sharp Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary, as well as the existence of the corresponding extremals. This generalizes early results of Chang-Yang…
We study divisors in the blow-up of $\mathbb{P}^n$ at points in general position that are non-special with respect to the notion of linear speciality introduced in [5]. We describe the cohomology groups of their strict transforms via the…
We report a new type of drop instability, where the density difference between the drop and the solvent is negative. We show that the drop falls inside the solvent down to a minimum height, then fragmentation takes place and secondary…
Stack-theoretic blow-ups have proven to be efficient in resolving singularities over fields of characteristic zero. In this article, we move forward towards positive characteristic where new challenges arise. In particular, the dimension of…
In this paper, we show how to extend the twin blow-up method recently developped by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive Hamilton-Jacobi equation posed in a…