Lower semicontinuity in $GSBD$ for nonautonomous surface integrals

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-04-11 DOI:10.1051/cocv/2023001

V. Cicco, G. Scilla

引用次数: 0

Abstract

We provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of $GSBD^p$ functions, whose dependence on the $x$-variable is $W^{1,1}$ or even $BV$: the notion of \emph{nonautonomous symmetric joint convexity}, which extends the analogous definition devised for autonomous integrands in \cite{FPS} where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in $SBV$ obtained in \cite{ACDD}, and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials.

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非自治曲面积分$GSBD$的下半连续性

我们提供了定义在$GSBD^p$函数空间上的非自治非强制表面能的下半连续性的一个充分条件，其对$x$变量的依赖为$W^{1,1}$甚至$BV$:\emph{非自治对称联合凸性}的概念，它扩展了在\cite{FPS}中假设近似向量场的保守性的自治积分的类似定义。这个条件可以推广到\cite{ACDD}中得到的一个在$SBV$中的非自治链式，这是证明下半连续性结果的一个关键工具。对于非均匀材料中断裂变分模型所产生的某些类型的表面能，这种新的节理凸性可以得到明确的检验。

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

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

自引率

7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.