On De Giorgi's conjecture of nonlocal approximations for free-discontinuity problems: The symmetric gradient case.

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2026-01-01 Epub Date: 2026-04-13 DOI:10.1007/s00526-026-03338-w

S Almi, E Davoli, A Kubin, E Tasso

引用次数: 0

Abstract

We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis is a crucial hurdle, which we overcome by generalizing a Fréchet-Kolmogorov approach previously introduced by two of the authors. A second essential difficulty is the identification of the limiting space of admissible deformations, since a control on the directional variations is, a priori, only available in average. A limiting representation in GSBD is eventually established via a novel characterization of this space.

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关于自由不连续问题的非局部近似的De Giorgi猜想：对称梯度情况。

我们证明了E. De Giorgi关于自由不连续问题的非局部近似的猜想推广到用可容许场的对称梯度定义的泛函的情况。在引入一类合适的连续有限差分近似后，我们证明了具有等界能量的变形的紧性及其收敛性。紧凑性分析是一个关键的障碍，我们通过推广之前由两位作者介绍的fr切特-科尔莫戈洛夫方法来克服它。第二个基本困难是可容许变形的极限空间的确定，因为对方向变化的控制是先验的，只能在平均情况下可用。GSBD中的极限表示最终通过对该空间的新颖表征建立起来。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.