We study variational problems involving nonlocal supremal functionals

where \(\Omega \subset \mathbb {R}^n\) is a bounded, open set and \(W:\mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}\) is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for \(L^\infty \)-weak\(^*\) lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. The analogous statement in the related context of double-integral functionals was recently shown to be false. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak\(^*\) closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.

我们研究涉及非局部超功能的变异问题

其中\（\ Omega \ subset \ mathbb {R} ^ n \）是有界的开放集，而\（W：\ mathbb {R} ^ m \ times \ mathbb {R} ^ m \ rightarrow \ mathbb {R} \）是合适的功能。根据直接存在方法的存在理论，我们确定了\（L ^ \ infty \）-弱\（^ * \）的充要条件这些功能的半连续性较低，即对称的和适当对角线形式的独立的水平凸度。更一般地说，我们表明在放松过程中保留了功能的最高结构。最近在双积分函数相关上下文中的类似陈述被证明是错误的。我们的证明主要依赖于最高功能和指标功能之间的联系。这使我们可以将松弛问题重塑为表征一类非局部包含物的弱\（^ * \）闭包的特征，这是具有独立意义的。为了说明该理论，我们为具有不同多孔要求的函数示例确定了显式松弛公式。