论文标题
由连续力学动机的有界域中的非局部梯度：微积分和嵌入的基本定理
Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings
论文作者
论文摘要
In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of interaction among particles, in the terminology of Peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel.因此，我们定义了适用于变化和部分微分方程的计算中的非局部模型的功能空间。我们的动机是开发适当的功能分析框架，以解决连续力学中的非本地模型，该模型需要与有限域一起工作，同时保留Riesz S-Fractional梯度的良好数学特性。该功能空间是用Sobolev和Bessel分数始终定义的：我们认为在自然规范下的平滑函数的闭合是该功能的LP规范及其非局部梯度的总和。在这项调查中的结果中，我们重点介绍了微积分基本定理的非本地版本（即，可以从其非局部梯度中恢复函数的表示公式），这使我们能够以Poincaré，Morrey，Trudinger，Trudinger，Trudinger和Hardy以及相应的紧凑型嵌入的精神证明不平等。这些结果足以显示出在凸度的假设下的一般能量功能的最小化。在这种非局部情况下，在这种非局部情况下的平衡条件也可以看作是有界域中的新的非局部偏微分方程。
In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of interaction among particles, in the terminology of Peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for nonlocal models in Calculus of Variations and partial differential equations. Our motivation is to develop the proper functional analysis framework in order to tackle nonlocal models in Continuum Mechanics, which requires working with bounded domains, while retaining the good mathematical properties of Riesz s-fractional gradients. This functional space is defined consistently with Sobolev and Bessel fractional ones: we consider the closure of smooth functions under the natural norm obtained as the sum of the Lp norms of the function and its nonlocal gradient. Among the results showed in this investigation we highlight a nonlocal version of the Fundamental Theorem of Calculus (namely, a representation formula where a function can be recovered from its nonlocal gradient), which allows us to prove inequalities in the spirit of Poincaré, Morrey, Trudinger and Hardy as well as the corresponding compact embeddings. These results are enough to show the existence of minimizers of general energy functionals under the assumption of convexity. Equilibrium conditions in this nonlocal situation are also established, and those can be viewed as a new class of nonlocal partial differential equations in bounded domains.
