单相自由边界问题的非平凡周期解

中文摘要 AI 辅助

我们在 $\mathbb{R}^3$ 中构造了单相 Bernoulli 自由边界问题的新经典全局解，其正相是 Hauswirth--Hélein--Pacard 双发夹域与一条直线的乘积的周期扰动。与先前对起伏圆柱的构造相比，基解的自由边界在模平移意义下不是紧的。更一般地，我们证明相同的方案在 $\mathbb{R}^{n+1}$（$n \geq 3$）中产生周期解，这些解从乘积解 $U(x,t)=u(x)$ 分岔，当基 $u:\mathbb{R}^n\to [0,\infty)$ 具有非平凡的有限 Morse 指标、在某个对称类内非退化，并满足无穷远处的某些自然几何条件时。在维数 $n\in\{3,4\}$ 中，由于稳定 Bernstein 定理可用，Fernández-Real、Florit-Simon 和 Serra 对有限 Morse 指标解的最新分类表明，每个这样的解都满足这些几何条件。因此，在 $\mathbb{R}^{4}$ 和 $\mathbb{R}^{5}$ 中从各自乘积分岔的起伏解的存在性归结为基解的非退化性质。为了确认扰动解是正的，我们证明了具有非紧自由边界的有限 Morse 指标解满足强距离非退化估计。

英文摘要

We construct new classical global solutions of the one-phase Bernoulli free boundary problem in $\mathbb{R}^3$ whose positive phases are periodic perturbations of the product of the Hauswirth--Hélein--Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in $\mathbb{R}^{n+1}$, $n \geq 3$, bifurcating from product solutions $U(x,t)=u(x)$, when the base $u:\mathbb{R}^n\to [0,\infty)$ has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions $n\in\{3,4\}$, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in $\mathbb{R}^{4}$ and $\mathbb{R}^{5}$ bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.

发表机构

• Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）

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48 pages