当$p$趋于1时$p$-调和函数边界导数的爆炸与衰减：非局部性
Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality
AI总结：
研究p趋于1时p-调和函数边界导数的爆炸与衰减，发现其由非局部数据决定，并给出临界圆柱例子。
摘要：
我们考虑在有界Lipschitz域$\Omega\subset\mathbb R^d$上$p$-Laplacian的Dirichlet问题，边界数据由闭集的指示函数给出。在数据消失的光滑边界点处，我们研究当$p\downarrow1$时上下内向法向Dini导数。我们给出了这些导数阶为$(p-1)^{-1}$的充分几何条件，以及它们关于$1/(p-1)$指数衰减的其他条件。爆炸或衰减是否发生并非由局部决定：在一个具有实解析边界的固定平面域上，在观测点均匀分离的任意小总长度集合上改变数据，会将爆炸变为指数衰减。我们还展示了一个在$\mathbb R^{d+1}$中圆柱体的临界例子，其中内向法向导数的阶为$\sqrt{d/(p-1)}$。
英文摘要：
We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $Ω\subset\mathbb R^d$, with boundary data given by the indicator of a closed set. At a smooth boundary point where the data vanish, we study the upper and lower inward normal Dini derivatives as $p\downarrow1$. We give sufficient geometric conditions for these derivatives to be of order $(p-1)^{-1}$, and other conditions under which they decay exponentially in $1/(p-1)$. Whether explosion or decay occurs is not determined locally: on one fixed planar domain with real-analytic boundary, changing the data on a set of arbitrarily small total length, uniformly separated from the observation point, changes explosion into exponential decay. We also exhibit a critical example of a cylinder in $\mathbb R^{d+1}$ where the inward normal derivative is of order $\sqrt{d/(p-1)}$.
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