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arXiv · 1202.0218

Asymptotic Behavior in Degenerate Parabolic Fully Nonlinear equations and its application to Elliptic Eigenvalue Problems

Soojung Kim·Ki-ahm Lee
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Abstract

We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+ μ\vp^{p}=0, \quad \vp>0\quad\text{in $Ω$} \vp&=0\quad\text{on $\pΩ$} posed in a (strictly) convex and smooth domain $Ω\subset\re^n$ for $0< p \leq 1,$ where $F(\cdot)$ is uniformly elliptic, positively homogeneous of order one and concave. We establish that $\log (\vp)$ is concave in the case $p=1$ and that the function $\vp^{\frac{1-p}{2}}$ is concave for $0<p<1.$

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Soojung Kim, Ki-ahm Lee. 2012-02-01. Asymptotic Behavior in Degenerate Parabolic Fully Nonlinear equations and its application to Elliptic Eigenvalue Problems. https://arxiv.org/abs/1202.0218

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