AI 中文总结
研究分数阶对数 $p$-拉普拉斯算子的正则性，采用迪乔吉 - 纳什 - 莫泽技术证明哈拿克不等式与局部赫尔德正则性，构造反例说明尾项对哈拿克不等式的必要性，结果在 $p = 2$ 时也新颖。
AI 中文摘要
我们证明了分数阶对数 $p$-拉普拉斯算子的哈拿克不等式（带尾项）和局部赫尔德正则性，该算子是通过对分数阶 $p$-拉普拉斯算子关于其阶数求导得到的。对于合适的函数 $u$，该算子在任意阶数 $s\in(0,1)$ 处可表示为一阶导数。其核包含一个对数因子，在大尺度上会改变符号，且在对角线附近比分数阶 $p$-拉普拉斯算子的核更奇异。我们采用经典的迪乔吉 - 纳什 - 莫泽技术来获得正则性估计。还构造了一个例子表明没有尾项时哈拿克不等式不成立。即使在 $p = 2$ 的线性情形下，我们的结果也是新的。
英文摘要
We prove the Harnack inequality (with tails) and local Hölder regularity for the fractional logarithmic $p$-Laplace operator, which is derived by differentiating the fractional $p$-Laplace operator with respect to its order. To be more precise, for a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-Δ_p)^{s+\log} u:= \frac{\rm d}{{\rm d}t}(-Δ_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order $s\in (0, 1).$ The kernel of this operator involves a logarithmic factor. As a consequence, it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional $p$-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in this setting. We also construct an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear setup $p=2$.
Comments51 pages. Comments are welcome!
