Quantitative uniqueness estimates for -Laplace type equations in the plane

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• 作者：Chang-Yu Guo^a ; ^b ; ^{changyu.guo@unifr.ch" class="auth_mail" title="E-mail the corresponding author} ; Manas Kar^a ; ^{manas.m.kar@jyu.fi" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Strong unique continuation principle ; pp-Laplace equation ; Beltrami equation
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：143
• 期：Complete
• 页码：19-44
• 全文大小：790 K

文摘

In this article our main concern is to prove the quantitative unique estimates for the p$p$-Laplace equation, 609dec5de17f8a47f6951913d3d8d" title="Click to view the MathML source">1<p<∞$1<p<\infty$, with a locally Lipschitz drift in the plane. To be more precise, let 609ba7b04d9a83e1bc584caea01">$u\in W_{loc}^{1,p}\left({\mathbb{R}}^2\right)$ be a nontrivial weak solution to where ecc3" title="Click to view the MathML source">W$W$ is a locally Lipschitz real vector satisfying ${\Vert W\Vert }_{L^q\left({\mathbb{R}}^2\right)}\le \tilde{M}$ for q≥max{p,2}$q\ge max\{p,2\}$. Assume that ec996a77a17bfa27ae3970607" title="Click to view the MathML source">u$u$ satisfies certain a priori assumption at 0. For q>max{p,2}$q>max\{p,2\}$ or q=p>2$q=p>2$, if ‖u‖_L^∞(R²)≤C₀${\Vert u\Vert }_{L^{\infty }\left({\mathbb{R}}^2\right)}\le C_0$, then ec996a77a17bfa27ae3970607" title="Click to view the MathML source">u$u$ satisfies the following asymptotic estimates at R≫1$R\gg 1$ where C>0$C>0$ depends only on p$p$, q$q$, 2153e444d7ba113b0a04347cb3ad4">$\tilde{M}$ and C₀$C_0$. When q=max{p,2}$q=max\{p,2\}$ and p∈(1,2]$p\in (1,2]$, if |u(z)|≤|z|^m$\left|u\left(z\right)\right|\le {\left|z\right|}^m$ for |z|>1$\left|z\right|>1$ with some m>0$m>0$, then we have where ecdd4895234e06766" title="Click to view the MathML source">C₁>0$C_1>0$ depends only on m,p$m,p$ and C₂>0$C_2>0$ depends on $m,p,\tilde{M}$. As an immediate consequence, we obtain the strong unique continuation principle (SUCP) for nontrivial solutions of this equation. We also prove the SUCP for the weighted p$p$-Laplace equation with a locally positive locally Lipschitz weight.