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Green representations and global Hölder continuity for solutions of elliptic equations

Duc Duong

Abstract

Let $N\in\mathbb{N}$ and $u$ be a weak solution of equation $\displaystyle Lu\equiv - \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{j}}(\frac{\partial u}{\partial x_{i}}b^{ij})= f$ in $Ω\subset \mathbb{R}^{N}$. We obtain functions $G$ and $H_{l}$ on $Ω\times Ω$ for every $l\in\{1,\cdots,N\}$ having following properties: if $f$ is in $L^{1}(Ω)$, then $\int_ΩG(x,y)f(x)dx = u(y)$, $\int_ΩH_{l}(x,y)f(x)dx = -\frac{\ \partial u}{\partial x_{l}}(y)\quad a.e~y\in Ω, \forall~l\in\{1,\cdots,N\}.$

Green representations and global Hölder continuity for solutions of elliptic equations

Abstract

Let and be a weak solution of equation in . We obtain functions and on for every having following properties: if is in , then ,
Paper Structure (6 sections, 13 theorems, 225 equations)

This paper contains 6 sections, 13 theorems, 225 equations.

Table of Contents

  1. Keywords
  2. MSC Classification
  3. Introduction
  4. Functions spaces
  5. Superpositions in $W_{B,A}(\Omega)$.
  6. Auxiliary functions

Key Result

Theorem 1

Let $A$ be admissible with respect to $\Omega$ and $\zeta \in (0,1)$. Then there are $t(\zeta)\in (\frac{2N^{2}+2N-2}{N^{2}+2N-1},2)$ and $\overline{r}(\zeta)\in (2,\frac{t(\zeta)(N+1)-2}{N-t(\zeta)})$ such that for every $t\in (t(\zeta),2)$, $r=\frac{t(N+1)-2}{N-t}$ and $\overline{r}\in (2,\overlin in two following cases $(i)$$N\in \{2,\cdots,8\}$ and c9 holds. $(ii)$$N\in \{3,4,\cdots\}$, $A=\

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Definition 3
  • ...and 18 more