论文标题
卷积型功能的变分理论
A variational theory of convolution-type functionals
论文作者
论文摘要
我们提供了一类用可集成核建模的功能类扰动的扰动处理，该功能近似于梯度的$ p $ th norm tong the narm narm the narm the bernel the infor the bernel的范围，通过让小参数$ \ varepsilon $倾向于0美元。我们首先提供了必要的功能分析工具，以在$ l^p $中显示强制性。主要结果是紧凑性和积分代理定理，该定理表明卷积型能量的限制是标准的本地积分功能，在Sobolev空间上定义了$ p $增长。该结果用于获得定期均质化结果，以研究针对点云上定义的功能，随机均质化和研究相关梯度流的限制的应用。
We provide a general treatment of perturbations of a class of functionals modeled on convolution energies with integrable kernel which approximate the $p$-th norm of the gradient as the kernel is scaled by letting a small parameter $\varepsilon$ tend to $0$. We first provide the necessary functional-analytic tools to show coerciveness in $L^p$. The main result is a compactness and integral-representation theorem which shows that limits of convolution-type energies is a standard local integral functional with $p$-growth defined on a Sobolev space. This result is applied to obtain periodic homogenization results, to study applications to functionals defined on point-clouds, to stochastic homogenization and to the study of limits of the related gradient flows.
