具有可积电荷密度的静电Born-Infeld方程
The electrostatic Born-Infeld equations with integrable charge densities
摘要：
本文研究了静电玻恩-因费尔德能量\begin{equation*} \int_{\mathbb{R}^n}1-\sqrt{1-|D v|^2}\ dx-\int_{\mathbb{R}^n}ρv\ dx, \end{equation*}在无穷远处为零的极小值。我们证明了极小元$u$是严格类空的，并且对于某些$p>n\geq 3$，只要$ρ$在解空间和$ρ\in L^p(\mathbb{R}^n)$的对偶空间中，它就是\begin{equation*}-\operatorname{div}\Big(\frac{D u}{\sqrt{1-|D u|^2}}\Big)=ρ,\end{equation*}的弱解.此外，我们还为某些$α\in (0,1)$提供了$u\in C^{1,α}(\mathbb{R}^n)$。
英文摘要：
We study the minimizer of the electrostatic Born--Infeld energy \begin{equation*} \int_{\mathbb{R}^n}1-\sqrt{1-|D v|^2}\ dx-\int_{\mathbb{R}^n}ρv\ dx, \end{equation*} which vanishes at infinity. We show that the minimizer $u$ is strictly spacelike and it is a weak solution to \begin{equation*}-\operatorname{div}\Big(\frac{D u}{\sqrt{1-|D u|^2}}\Big)=ρ,\end{equation*} provided that $ρ$ is in the dual space of the solution space and $ρ\in L^p(\mathbb{R}^n)$, for some $p>n\geq 3$. Moreover, we have $u\in C^{1,α}(\mathbb{R}^n)$ for some $α\in (0,1)$.
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