Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.33900

A remark on the global minimality of the cracktip

Francesco Deangelis·Camillo De Lellis·Matteo Focardi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a short proof that the planar cracktip is a global minimizer of the Mumford--Shah functional, based on interior and Dirichlet boundary regularity. We construct a Dirichlet minimizer with nonempty discontinuity set and show that this set contains no simple closed curves. The regularity results then imply the existence of an interior regular loose end, at which a blow-up yields the cracktip.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Francesco Deangelis, Camillo De Lellis, Matteo Focardi. 2026-10-24. A remark on the global minimality of the cracktip. https://arxiv.org/abs/2610.33900

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Bennati-Dragulescu-Yakovenko model in the continuous setting: PDE derivation and long-time behavior

In this manuscript, we develop and analyze a continuous version of the well-known Bennati-Dragulescu-Yakovenko (BDY) dollar-exchange discrete model. Starting from the conservative BDY exchange mechanism, we rely on kinetic theory for multi-agent systems in order to propose an analogous Boltzmann-type continuous dynamics, which does not belong to the class of other popular kinetic models for wealth exchange. We employ the quasi-invariant limit procedure to rigorously derive a nonlinear PDE on the half-line, which is a Fokker-Planck equation featuring the boundary value in the drift term. The PDE is supplemented with a nonlinear Robin-type boundary condition encoding conservation of total agents and wealth. We prove relative entropy decay to the unique stationary state that is the Boltzmann-Gibbs (exponential) distribution with a $\mathrm{e}^{-c\sqrt{t}}$ convergence rate. We determine a new entropy production term which is a Fisher information minus a competing term which measures the temporal distance of the boundary term from its equilibrium value. Thanks to an inequality relating the Fisher information relative to the Boltzmann-Gibbs equilibrium to the $L^1$-norm, we prove $L^1$-convergence toward equilibrium and convergence of the boundary term (up to a subsequence), as well as convergence in both the Hellinger and Fourier distances. This represents an advancement in classical arguments of entropy decay for Fokker-Planck equations in a kinetic framework.

math.AP↗
arXiv2026

On the Dirichlet boundary value problem on Cartan-Hadamard manifolds

We study bounded solutions of Dirichlet boundary value problems for semilinear elliptic equations on Cartan-Hadamard manifolds with pinched negative curvature. We prove nonexistence when the asymptotic boundary has zero Hausdorff measure at an exponent determined by the upper curvature bound and the spectral parameter. The result holds for continuous viscosity solutions under a one-sided bound on the nonlinearity, including the Allen-Cahn equation. The proof combines convex barriers at infinity, distance comparison for $C^{1,1}$ hypersurfaces and a spectral maximum principle. For surfaces, no lower curvature bound is needed.

math.AP↗
arXiv2026

Improved well-posedness for the limit flow of differentiation of roots of polynomials

In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberger \cite{steinerberger2019nonlocal} on the real line and which represents the evolution of the density of the roots of polynomials under differentiation. After integrating the partial differential equation in question, we observe that it can be treated with the theory of viscosity solutions. This equation at hand is a non linear parabolic integro-differential equation which involves the elliptic operator called the half-Laplacian. Due to the singularity of the equation, we first restrict our study to strictly positive initial condition. We obtain a comparison principle for solutions of the primitive equation which yields uniqueness, existence, continuity with respect to initial condition. We then extend these results to general initial data. We finally prove that the system of particles on the circle introduced by Steinerberger indeed approximates the solution of the equation.

math.AP↗