连续介质力学中有界区域的非局部梯度：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
摘要：
在本文中，我们开发了一套新的结果的基础上的非局部梯度联合启发的Riesz s-分数梯度和Peridendics，在这个意义上说，它的积分域取决于一个球的半径δ> 0（地平线的粒子之间的相互作用，在Peridendics的术语），同时保持在同一时间的奇异性的里斯位势在其积分内核。相应地，我们定义了一个适用于变分法和偏微分方程中的非局部模型的函数空间。我们的动机是开发适当的泛函分析框架，以解决连续介质力学中的非局部模型，这需要使用有界域，同时保留Riesz s-分数阶梯度的良好数学性质。该函数空间的定义与Sobolev和Bessel分式空间一致：我们考虑光滑函数在自然范数下的封闭性，该自然范数是函数及其非局部梯度的Lp范数之和。在这次调查中显示的结果中，我们强调了微积分基本定理的非局部版本（即，一个函数可以从其非局部梯度中恢复的表示公式），这使我们能够证明Poincaré，Morrey，Trudinger和Hardy的精神中的不等式以及相应的紧嵌入。这些结果足以证明在凸性假设下一般能量泛函极小解的存在性。建立了这种非局部情形下的平衡条件，这些平衡条件可以看作是有界区域上一类新的非局部偏微分方程。
英文摘要：
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. 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.
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