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

Abstract

We study the asymptotic behavior of the nonlinear parabolic flows $u_t=F(D^2{u}^m)$ when $t\ra \infty$ for $m\ge 1$, 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 $\Omega$} \vp&=0\quad\text{on $\pΩ$} posed in a (strictly) convex and smooth domain $Ω\subset\re^n$ for $0<p\le 1,$ 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 $0<p<1.$

References in corpus (1)

• Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations