NDLI: On the entire self-shrinking solutions to Lagrangian mean curvature flow

Content Provider	SpringerLink
Author	Huang, Rongli Wang, Zhizhang
Copyright Year	2010
Abstract	The authors prove that the logarithmic Monge–Ampère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time t = 0. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation $$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$ should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function \|x\|2 D 2 u at infinity has an uniform positive lower bound larger than 2(1 − 1/n). Using a similar method, we can prove that every classical convex or concave solution of the equation $$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$ must be a quadratic polynomial, where λi are the eigenvalues of the Hessian D 2 u.
Ending Page	339
Page Count	19
Starting Page	321
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3-4
Volume Number	41
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-08-17
Publisher Place	Berlin, Heidelberg
Access Restriction	Subscribed
Subject Keyword	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Minimal surfaces, surfaces with prescribed mean curvature
Content Type	Text
Resource Type	Article
Subject	Analysis Applied Mathematics

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