论文标题
笛卡尔凸度是非局部至上功能的变分存在理论中的关键概念
Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals
论文作者
论文摘要
由$ l^{\ infty} $变化的直接方法的动机，我们的主要结果确定了凸的概念，这些概念表征了弱$^*$较低的非局部至上函数的较低的半持续性：笛卡尔级别凸度。这个新概念与一维环境中的单独级别凸度相吻合，对于更高的维度而言，它严格较弱。我们讨论了矢量案例中的放松，表明放松的功能通常不会保持至上的形式。除了用多孔类型的例子说明这一事实外，我们还提供了结构保护的精确标准。保留结构后，将根据（对角）原始载体的笛卡尔级凸构来给出表示公式。这项工作不仅完成了[Kreisbeck \＆Zappale，calc。〜var。〜pde，2020]中启动的分析图，而且还与双重积分建立了联系。我们通过$ l^p $ -Approximation从$γ$ -Convergence中使用两类功能，用于分化集成性指数。这些证据利用了非局部包含及其渐近行为的最新结果，并使用了年轻的测量理论和凸分析中的工具。
Motivated by the direct method in the calculus of variations in $L^{\infty}$, our main result identifies the notion of convexity characterizing the weakly$^*$ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in [Kreisbeck \& Zappale, Calc.~Var.~PDE, 2020], but also establishes a connection with double integrals. We relate the two classes of functionals via an $L^p$-approximation in the sense of $Γ$-convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis.
