Mathematics – Analysis of PDEs
Scientific paper
2009-02-12
Mathematics
Analysis of PDEs
73pages
Scientific paper
We consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $\Delta u + f(u) = 0 \hbox{in} \R^3 $. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. We assume that $M$ has $m\ge 2$ ends, and additionally that $M$ is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman-Meeks surface of any genus). We prove that for any small $\alpha >0$, the Allen-Cahn equation has a family of bounded solutions depending on $m-1$ parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface $M_\alpha := \alpha^{-1} M$, with ends possibly diverging logarithmically from $M_\A$. We prove that these solutions are $L^\infty$-{\em non-degenerate} up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.
Kowalczyk Mike
Pino Manuel del
Wei Juncheng
No associations
LandOfFree
Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in $\R^3$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in $\R^3$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Entire Solutions of the Allen-Cahn equation and Complete Embedded Minimal Surfaces of Finite Total Curvature in $\R^3$ will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-556037