Asymptotic behavior in degenerate parabolic fully nonlinear equations and its application to elliptic eigenvalue problems
2014.07.23 17:18
| 실적년도 | 2013년 |
|---|---|
| 논문구분 | 국외 |
| 총저자 | Soojung Kim, Ki-Ahm Lee |
| 학술지명 | Journal of Differential Equations |
| 권(Vol.) | 254 |
| 호(No.) | 8 |
| 게재년월 | 2013년 4월 |
| Impact Factor | |
| SCI 등재 | SCI |
| 비고 |
We study the fully nonlinear parabolic equation. F(D2um)-ut=0 in Ω×(0,+∞),m≥1, with the Dirichlet boundary condition and positive initial data in a smooth bounded domain Ω⊂Rn, provided that the operator F is uniformly elliptic and positively homogeneous of order one. We prove that the renormalized limit of parabolic flow u(x, t) as t→. +. ∞ is the corresponding positive eigenfunction which solves. F(D2φ)+μφp=0 in Ω, where 0<p:=1/m ≤ 1 and μ > 0 is the corresponding eigenvalue. We also show that some geometric property of the positive initial data is preserved by the parabolic flow, under the additional assumptions that Ω is convex and F is concave. As a consequence, the positive eigenfunction has such geometric property, that is, log(φ) is concave in the case p=1, and φ 1-p/2 is concave for 0 < p< 1.
