Perron极值解在Bernoulli单相问题中的变分性质
Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem
- Lafayette College（拉斐特学院）
- University of Utah（犹他大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究Bernoulli单相问题的Perron极值解，证明其具有内变分性质，并刻画二维情形下奇点结构：最大下解边界光滑，最小上解边界点或光滑或为等斜率两平面楔形。
AI中文摘要：
我们研究平稳单相Bernoulli问题的Perron极值解。这些解在多个应用中很重要，但由于其非变分构造，关于其自由边界正则性的已知结果不多。我们证明，事实上，Perron极值解保留了一些变分特征；具体而言，它们是内变分意义下的解。我们通过证明极值解作为抛物型Bernoulli问题的单调解的无限时间极限出现来确立这一性质。作为推论，我们能够描述二维情况下Perron极值解的奇点结构：最大下解具有光滑的自由边界，而最小上解的自由边界点要么是光滑的，要么其爆破极限是斜率相等的两平面楔形。
英文摘要：
We study the Perron extremal solutions of the stationary one-phase Bernoulli problem. These solutions are important in several applications, but not much is known about the regularity of their free boundaries due to their non-variational construction. We show that, in fact, Perron extremal solutions retain some variational features; specifically, they are solutions in the sense of inner variations. We establish this property by showing that extremal solutions arise as infinite-time limits of monotone solutions of the parabolic Bernoulli problem. As a consequence, we are able to describe the structure of singularities for Perron extremal solutions in two dimensions: largest subsolutions have smooth free boundaries, while smallest supersolutions have free boundary points that are either smooth, or whose blow-up limits are two-plane wedges with equal slope.
