论文标题
具有可整合电荷密度的静电出生式进程方程
The electrostatic Born-Infeld equations with integrable charge densities
论文作者
论文摘要
我们研究静电出生的最小化器 - infeld Energy \ begin {equation*} \ int _ {\ Mathbb {r}^n} 1- \ sqrt {1- \ sqrt {1- | d v |^2} \ dx- \ dx- \ dx- \ int _ {无穷大。我们表明，最小化的$ u $严格存在空间，它是\ begin {equination*} - \ operatorName {div} \ big（\ frac {\ frac {\ sqrt {\ sqrt {\ sqrt {1- | d u | d u |^2}}} \ big）= extept un的解决方案* $ρ\ in l^p（\ mathbb {r}^n）$，对于某些$ p> n \ geq 3 $。此外，我们在c^{1，α}（\ mathbb {r}^n）$中有$ u \，对于某些$α\ in（0,1）$。
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 $α\in (0,1)$.
