Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Local Solutions of Second Order Quasi-linear Elliptic Systems with Arbitrary 1-Jet at a Point

Yifei Pan, Yu Yan

Abstract

We prove a general result on the existence of local solutions of any second order quasi-linear elliptic system with arbitrary 1-jet at a point.

On Local Solutions of Second Order Quasi-linear Elliptic Systems with Arbitrary 1-Jet at a Point

Abstract

We prove a general result on the existence of local solutions of any second order quasi-linear elliptic system with arbitrary 1-jet at a point.
Paper Structure (8 sections, 10 theorems, 131 equations)

This paper contains 8 sections, 10 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Defining the Function Spaces and Some Preliminary Estimates
  3. A Hölder Estimate for the Newtonian Potential
  4. The Integral System and the Map
  5. The Estimate for $\|\bm{\Theta}(\bm{f} )\|^{(2, \alpha)}$
  6. Estimates for $\|A_j\|_{\alpha}$, $\|B_k^j\|_{\alpha}$, $\|C^j_{kl}\|_{\alpha}$, and $\|E_j\|_{\alpha}$
  7. The Estimates for $\| \bm{\Theta} (\bm{f}) \|^{(2, \alpha)}$
  8. The Estimates for $\| \bm{\Theta} (\bm{f}) - \bm{\Theta} (\bm{g}) \|^{(2, \alpha)}$

Key Result

Theorem 1.1

Let $\bm{\phi} (x, p, q)= (\phi ^1(x, p, q), ..., \phi ^m(x, p, q) ):B_R \times B'_{R'} \times \mathbb{R}^{mn} \to \mathbb{R}^{m}$ be of $C_{loc}^{1,\alpha}$$(0 < \alpha < 1)$. For any given $\bm{c_0} \in B'_{R'}$ and $\bm{c_1}\in \mathbb{R}^{mn}$, the elliptic quasi-linear system has $C^{2,\alpha}$ solutions $\bm{u}(x)$ from $B_R$ to $B'_{R'}$ when $R$ is sufficiently small.

Theorems & Definitions (11)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • ...and 1 more