具有孤立临界点的整体 $p$-调和函数及自然梯度的 $C^1$-正则性失效
Entire $p$-harmonic functions with an isolated critical point and failure of $C^1$-regularity of the natural gradient
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中文总结 AI 辅助
本文构造了具有唯一孤立临界点的齐次整体p-调和函数，证明在n≥3且p接近2时自然梯度非C^1正则，从而否定Balci-Diening-Weimar猜想的标量情形。
中文摘要 AI 辅助
对于每个 $n\ge2$ 和 $1<p<\infty$，我们证明了存在一个非常数的齐次整体 $p$-调和函数，其唯一临界点是原点。这些函数的角部关于赤道轴对称且为偶函数。为建立该解，我们在极点处使用压缩论证并结合射击论证。由此可知，在每个维度 $n\ge3$ 且 $p<2$ 充分接近 $2$ 时，自然梯度在原点处的逐点 Hölder 指数严格小于 1。这些例子通过反驳 $C^1$ 断言和线性 $L^2$ 平均振荡估计，解决了 Balci、Diening 和 Weimar 猜想的剩余标量情形。
英文摘要
For every $n\ge2$ and $1<p<\infty$, we show an existence of a nonconstant homogeneous entire $p$-harmonic function whose only critical point is the origin. The angular parts of these functions are axially symmetric and even across the equator. To establish the solution we use contraction argument at the pole in conjuction with a shooting argument. It follows that in every dimension $n\ge3$ and for $p<2$ sufficiently close to $2$, the natural gradient has pointwise Hölder exponent strictly below one at the origin. Such examples settle the remaining scalar case of the conjecture of Balci, Diening, and Weimar by disproving both the $C^1$ assertion and the linear $L^2$ mean oscillation estimate.
