Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the electrostatic Born--Infeld equation with radial charge density

Nguyen The Cang

Abstract

In this note, we will give a new proof of the solvability of the electrostatic Born--Infeld equation with radial charge by using the conformal method and the Spacetime Positive Energy Theorem.

A note on the electrostatic Born--Infeld equation with radial charge density

Abstract

In this note, we will give a new proof of the solvability of the electrostatic Born--Infeld equation with radial charge by using the conformal method and the Spacetime Positive Energy Theorem.

Paper Structure

This paper contains 10 sections, 8 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal method
  4. Elliptic operators on weighted Hölder spaces
  5. Spacetime Positive Energy Theorem
  6. Proof of Theorem \ref{['main theorem']}
  7. Behavior of radial solutions to the conformal equation
  8. Existence of constraint solutions with prescribed radial mean curvature
  9. Vanishing ADM mass and spacelike hypersurfaces in $\mathbb L^{n+1}$
  10. Further discussion

Key Result

Theorem 1.1

Let $\rho$ be an arbitrary radial function in $C^{1, \alpha}_{ - q} (\mathbb R^n)$ with $\alpha \in (0,1)$ and Then there exists an asymptotically flat spake-like hypersurface $(\Sigma, h)$ of the Lorentz-Minkowski spacetime $\mathbb L^{n+1}$ with mean curvature $\rho$ and $h - \delta_{\text{Euc}} \in C^{3,\alpha}_{-2q + 2}$. In particular, the electrostatic Born-Infeld equation BI admits at leas

Theorems & Definitions (11)

  • Theorem 1.1
  • Proposition 2.1: Compact embedding for weighted Hölder spaces
  • Proposition 2.2: Weighted elliptic regularity for Laplacian
  • Proposition 2.3: Weighted elliptic regularity for vector Laplacian
  • Theorem 2.4: Existence and uniqueness of solution to the Lichnerowicz equation
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 1 more