奇异 Born-Infeld 型泛函的极小值点与弱解
Minimizers and Weak Solutions for Singular Born--Infeld Type Functionals
AI总结：
研究 Born-Infeld 型奇异泛函极小值点与弱解关系，提出单调逼近法处理奇异性，证明梯度不触奇异边界，在特定条件下建立正则性并给出估计，扩展经典理论提供统一框架。
AI中文摘要：
我们研究了一类源于 Born-Infeld 型理论$\mathcal{L}(s)$的奇异泛函的极小值点与弱解之间的关系。在静电场$s = \frac{1}{2}|\nabla\phi|^2$的设定下，$\mathcal{L}(s)$满足$\lim_{s\to(1/2)^-}\mathcal{L}(s)=+\infty$，这自然地强制了有限梯度界$|\nabla\phi|\leq1$，即截断阈值。对于规定的扩展电荷密度$\rho$，我们考虑系统的弱解与奇异泛函的极小值点$\phi_0$之间的关系。我们提出了一种单调逼近方法来处理$\mathcal{L}(s)$的内在奇异性。我们证明了极小值点的梯度永远不会触及奇异边界$|\nabla\phi|^2 = 1$；这个结构结果产生了一个关键的可积性性质、极小值点的存在性和唯一性以及相应的变分不等式。在$\rho$是径向分布的额外假设下，我们表明极小值点是唯一的弱解。此外，我们在对$\rho$的适当可积性条件下建立了极小值点的$C^1$和$C^2$正则性，并为严格类空条件$|\nabla\phi_0|\leq1 - \epsilon$提供了一个统一估计，其中参数$\epsilon>0$根据空间维度、空间区域和$\rho$明确表征。我们的结果将经典的 Born-Infeld 理论扩展到了一类一般的奇异 Born-Infeld 型理论，从而为这种奇异泛函和系统的变分分析和正则性提供了一个统一框架。
英文摘要：
We investigate the relation between minimizers and weak solutions for a class of singular functionals arising from Born--Infeld type theories $\mathcal{L}(s)$. In the setting of an electrostatic field $s=\frac{1}{2}|\nablaϕ|^2$, $\mathcal{L}(s)$ satisfies $\lim_{s\to(1/2)^-}\mathcal{L}(s)=+\infty$, which naturally enforces the finite gradient bound $|\nablaϕ|\le 1$, also called the truncation threshold. For a prescribed extended charge density $ρ$, we consider the relation between the weak solution of the system \begin{equation} \begin{cases} -{\rm div}\left(b\left(\frac12|\nablaϕ|^2\right)\nablaϕ\right)=ρ,& \text{in }\mathbb{R}^N,\\ b(s)=\mathcal{L}'(s),\quad\lim_{s\to\frac12^-} b(s)=+\infty,\\ \lim_{|x|\to\infty}ϕ(x)=0 \end{cases} \end{equation} and the minimizer $ϕ_0$ of the singular functional. We propose a monotonic approximation method to handle the intrinsic singularities of $\mathcal{L}(s)$. We prove that the gradient of the minimizer never touches the singular boundary $|\nablaϕ|^2=1$; this structural result yields a key integrability property, the existence and uniqueness of the minimizer, and the corresponding variational inequality. Under the additional assumption that $ρ$ is radially distributed, we show that the minimizer is the unique weak solution. Furthermore, we establish the $C^1$ and $C^2$ regularity of the minimizer under suitable integrability conditions on $ρ$, and provide a uniform estimate for the strict spacelikeness condition $|\nablaϕ_0|\le 1-ε$, where the parameter $ε>0$ is explicitly characterized in terms of the spatial dimension, the spatial region, and $ρ$. Our results extend the classical Born--Infeld theory to a general class of singular Born--Infeld type theories, thereby providing a unified framework for the variational analysis and regularity of such singular functionals and systems.
