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

TL;DR

This paper investigates the long-term behavior of certain nonlinear parabolic equations and explores geometric properties of solutions to related elliptic eigenvalue problems, revealing concavity properties of solutions under specific conditions.

Contribution

It establishes new concavity results for solutions of elliptic eigenvalue problems involving fully nonlinear operators, extending understanding of their geometric properties.

Findings

Logarithm of solutions is concave when p=1.

The power transformation of solutions is concave for 0<p<1.

Asymptotic analysis of nonlinear parabolic flows.

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) &+ \mu\vp^{p}=0, \quad \vp>0\quad\text{in $\Omega$} \vp&=0\quad\text{on $\p\Omega$} posed in a (strictly) convex and smooth domain $\Omega\subset\re^n$ 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 $0<p<1.$