发表机构
Fahad Bin Sultan University; Universidade Federal de São Carlos – UFSCar; Sapienza Università di Roma; Universidad Torcuato di Tella（法赫德国王苏丹大学; 圣卡洛斯联邦大学; 罗马第一大学; 托尔夸托·迪泰拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究无穷多个 p-Laplacian 叠加的 Dirichlet 问题，通过饱和通量刻画梯度约束下的可解性，并应用于 Born-Infeld 模型，获得电荷密度等周界和弱场展开收敛速率。
AI 中文摘要
我们研究无穷级数 $p$-Laplacian 的 Dirichlet 问题：$$-\sum_{p=2}^{\infty}a_{p}\Delta_{p}u=f \\ \text{ in } \Omega, \qquad u=g \\ \text{ on } \partial\Omega,$$ 其中 $\{a_{p}\}$ 是非负数列，其幂级数的收敛半径为 $\sigma\in(0,\infty]$。该算子形式为 $-\operatorname{div}(A(|\nabla u|)\nabla u)$，其中 $A$ 在 $|\nabla u|=\sigma$ 处奇异；模型情形包括 Minkowski 空间中的平均曲率算子和非线性静电学的 Born-Infeld 算子。收敛半径强制梯度约束 $\\|\nabla u\\|_\infty\le\sigma$，因此自然变分问题是有约束的。对于与约束相容的边界数据，相关能量的唯一极小元总是存在，并且总是满足变分不等式；我们确定饱和通量 $\Lambda:=\sum_{p\ge2}a_p\sigma^{p-1}$ 为控制方程可解性的量：若 $\Lambda<\infty$，则弱解存在仅当对每个有限周长集 $E\Subset\Omega$ 有 $|\int_E f|\le\Lambda P(E)$；因此对于 $f\equiv\lambda$，一旦 $\lambda>\Lambda h(\Omega)$（其中 $h(\Omega)$ 是 $\Omega$ 的 Cheeger 常数），则无解。该阈值在球上是尖锐的，此时极小元具有正测度的饱和区域 $\{|\nabla u|=\sigma\}$。当 $\Lambda=\infty$ 时（如 Born-Infeld 型算子），不存在此类障碍，且只要存在低于 $\sigma$ 的 Lipschitz 界，极小元就求解方程；对于 $f\equiv0$，在边界数据的有限斜率条件下，我们获得这样的界，且该界对级数的截断一致。作为应用，我们获得了具有饱和位移的非线性静电学所支持的电荷密度的等周界，以及 Born-Infeld 模型弱场展开的定量收敛速率。
英文摘要
We study the Dirichlet problem for an infinite series of $p$-Laplacians, $$-\sum_{p=2}^{\infty}a_{p}Δ_{p}u=f \ \text{ in } Ω, \qquad u=g \ \text{ on } \partialΩ,$$ where $\{a_{p}\}$ is a sequence of nonnegative numbers whose power series has radius of convergence $σ\in(0,\infty]$. The operator is formally $-\operatorname{div}( {A}(|\nabla u|)\nabla u)$ with $ {A}$ singular at $|\nabla u|=σ$; model cases are the mean curvature operator in Minkowski space and the Born-Infeld operator of nonlinear electrostatics. The radius of convergence forces the gradient constraint $\|\nabla u\|_\infty\leσ$, so the natural variational problem is constrained. A unique minimizer of the associated energy always exists (for boundary data compatible with the constraint) and always solves a variational inequality, and we identify the saturation flux $Λ:=\sum_{p\ge2}a_pσ^{p-1}$ as the quantity governing solvability of the equation: if $Λ<\infty$ a weak solution exists only if $|\int_E f|\leΛP(E)$ for every set of finite perimeter $E\SubsetΩ$, so for $f\equivλ$ no solution exists once $λ>Λh(Ω)$, $h(Ω)$ the Cheeger constant of $Ω$. The threshold is sharp on balls, where the minimizer has a saturation region $\{|\nabla u|=σ\}$ of positive measure. When $Λ=\infty$, as for Born-Infeld type operators, no such obstruction is present, and the minimizer solves the equation whenever a Lipschitz bound below $σ$ is available; for $f\equiv0$ we obtain such a bound, uniform in the truncations of the series, under a bounded slope condition on the boundary datum. As applications we obtain an isoperimetric bound on the charge densities supported by a nonlinear electrostatics with saturating displacement, and a quantitative convergence rate for the weak field expansion of the Born-Infeld model.
