整体 Sobolev 不等式与退化 p-Laplacian 方程
偏微分方程分析
2018-01-30 v1
摘要
我们证明了一个局部的弱 Sobolev 不等式利用一族 p p p Ω ⊂ R n \Omega\subset\mathbb{R}^n Ω ⊂ R n ρ \rho ρ Ω \Omega Ω Q Q Q Ω \Omega Ω n × n n\times n n × n Θ ⋐ Ω \Theta\Subset\Omega Θ ⋐ Ω ( \fint B ∣ f ∣ p σ d x ) 1 p σ ≤ C [ r ( B ) \fint B ∣ Q ∇ f ∣ p d x + \fint B ∣ f ∣ p d x ] 1 p \Big(\fint_B |f|^{p\sigma}dx\Big)^\frac{1}{p\sigma} \leq C\Big[ r(B)\fint_B |\sqrt{Q}\nabla f|^pdx + \fint_B |f|^pdx\Big]^\frac{1}{p} ( \fint B ∣ f ∣ p σ d x ) p σ 1 ≤ C [ r ( B ) \fint B ∣ Q ∇ f ∣ p d x + \fint B ∣ f ∣ p d x ] p 1 σ > 1 \sigma>1 σ > 1 B ⊂ Θ B\subset \Theta B ⊂ Θ f ∈ L i p 0 ( Θ ) f\in Lip_0(\Theta) f ∈ L i p 0 ( Θ ) ( ∫ Θ ∣ f ∣ p σ d x ) 1 p σ ≤ C ( ∫ Θ ∣ Q ∇ f ( x ) ∣ p d x ) 1 p \Big(\int_\Theta |f|^{p\sigma}dx\Big)^\frac{1}{p\sigma} \leq C\Big(\int_\Theta |\sqrt{Q}\nabla f(x)|^pdx\Big)^\frac{1}{p} ( ∫ Θ ∣ f ∣ p σ d x ) p σ 1 ≤ C ( ∫ Θ ∣ Q ∇ f ( x ) ∣ p d x ) p 1 { \mx p , τ u = φ in Θ u = 0 in ∂ Θ , \begin{cases} \mx_{p,\tau} u & = \varphi \text{in} \Theta \\ u & = 0 \text{in} \partial \Theta, \end{cases} { \mx p , τ u u = φ in Θ = 0 in ∂ Θ , \mx p , τ \mx_{p,\tau} \mx p , τ p p p \mx p , τ u = div ( ∣ Q ∇ u ∣ p − 2 Q ∇ u ) − τ ∣ u ∣ p − 2 u . \mx_{p,\tau} u = \text{div}\Big(\big|\sqrt{Q} \nabla u\big|^{p-2}Q\nabla u\Big) - \tau |u|^{p-2}u. \mx p , τ u = div ( Q ∇ u p − 2 Q ∇ u ) − τ ∣ u ∣ p − 2 u .
引用
@article{arxiv.1801.09610,
title = {Global Sobolev inequalities and Degenerate P-Laplacian equations},
author = {David Cruz-Uribe and Scott Rodney and Emily Rosta},
journal= {arXiv preprint arXiv:1801.09610},
year = {2018}
}