$p$-torsion函数到距离函数的收敛速率

中文摘要 AI 辅助

我们证明，在有界域上，Dirichlet $p$-torsion函数 $u_p$ 在一致范数下以 $O(1/p)$ 的速率收敛到到边界的距离函数 $d$，其中的常数仅依赖于维数和直径。上界估计来自一个极点在边界点的径向障碍函数，该函数在 $p>n$ 时是允许的比较函数，且不需要边界正则性；下界估计来自内切球。在球上，误差等于 $(1+\log n)/p+O(p^{-2})$，因此 $1/p$ 的阶在一般情况下无法改进。在一致外部球条件下，环形障碍函数给出对每个 $p>1$ 有 $u_p\le K_\Omega^{1/(p-1)}d$，其中 $K_\Omega$ 显式依赖于维数、直径和外部半径；在凸集上，常数为直径，并且当 $-\Delta d$ 具有正的分布意义下界时，$d$ 的倍数是一个上解。在 $C^2$ 域上，同样的障碍函数和经典梯度最大值原理给出 $\norm{\nabla u_p}_{L^\infty}^{p-1}\le K_\Omega$。当 $-\Delta d$ 是一个全变差有限的测度时（我们证明这在具有一致外部球的 $C^1$ 域上成立），梯度在 $L^q$ 中以 $O(p^{-1/2})$（对 $q\le2$）和 $O(p^{-1/q})$（对 $q\ge2$）的速率收敛；这些指数不声称是最优的。显式的球轮廓表明梯度不一致收敛。

英文摘要

We prove that the Dirichlet $p$-torsion function $u_p$ converges to the distance to the boundary $d$ at the rate $O(1/p)$ in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for $p>n$ and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals $(1+\log n)/p+O(p^{-2})$, so the order $1/p$ cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives $u_p\le K_Ω^{1/(p-1)}d$ for every $p>1$, with $K_Ω$ explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of $d$ is a supersolution whenever $-Δd$ has a positive distributional lower bound. On $C^2$ domains, the same barrier and the classical gradient maximum principle give $\norm{\nabla u_p}_{L^\infty}^{p-1}\le K_Ω$. When $-Δd$ is a measure of finite total variation, which we prove for $C^1$ domains with a uniform exterior ball, the gradients converge in $L^q$ at the rate $O(p^{-1/2})$ for $q\le2$ and $O(p^{-1/q})$ for $q\ge2$; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.

发表机构

• New Uzbekistan University（新乌兹别克斯坦大学）
• Sharif University of Technology（谢里夫理工大学）
• School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究院数学学院）

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