标题： 哈密顿-雅可比方程在自旋玻璃中的包络表示 显示英文标题

标题： Envelope presentation of Hamilton-Jacobi equations from spin glasses

摘要： 最近，[arXiv:2311.08980] 证明了，如果存在的话，可能非凸的自旋玻璃模型的极限自由能必须由相关无限维非凸哈密顿-雅可比方程的一个特征决定。 在本工作中，我们纯粹从偏微分方程（PDE）的角度研究了类似的主题。 具体来说，我们研究了上述方程的唯一粘性解，并推导出该解的一个包络型表示公式，形式由 Evans 在 [doi:10.1007/s00526-013-0635-3] 中提出。 解的值表示为沿特征线的值的平均值，权重为一个非显式的概率测度。 技术上的挑战不仅来自于无限维性，还来自于该方程定义在一个内部为空的闭凸锥上，而不是在整个空间上。 在引言中，我们提供了来自自旋玻璃理论的动机描述，并给出了相应的结果以与 PDE 结果进行比较。 显示英文摘要

摘要： Recently, [arXiv:2311.08980] demonstrated that, if it exists, the limit free energy of possibly non-convex spin glass models must be determined by a characteristic of the associated infinite-dimensional non-convex Hamilton-Jacobi equation. In this work, we investigate a similar theme purely from the perspective of PDEs. Specifically, we study the unique viscosity solution of the aforementioned equation and derive an envelope-type representation formula for the solution, in the form proposed by Evans in [doi:10.1007/s00526-013-0635-3]. The value of the solution is expressed as an average of the values along characteristic lines, weighted by a non-explicit probability measure. The technical challenges arise not only from the infinite dimensionality but also from the fact that the equation is defined on a closed convex cone with an empty interior, rather than on the entire space. In the introduction, we provide a description of the motivation from spin glass theory and present the corresponding results for comparison with the PDE results.