Some qualitative properties for geometric flows and its Euler implicit discretization. (May 2016)

• Record Type: Journal Article
• Authors: Alvarez, Luis; Díaz, Gregorio; Díaz, Jesús Ildefonso
• Is Part Of: Nonlinear analysis. Volume 137(2016)
• Page Start: 43
• Page End: 76
• Publication Date: 2016-05
• Subjects: 35K65 -- 35J60 -- 53C44 -- 35K55 -- 35L60 -- 35L81; Geometric flow -- Implicit discretization -- Nonlinear parabolic and elliptic equations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.11.023
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the geometric flow parabolic equation and its implicit discretization which yield a family of nonlinear elliptic problems. We show that there are important differences in the study of those equations which concerns the propagation of level sets of data. Our study is based on the previous study of radially symmetric solutions of the corresponding equation. Curiously, in radial coordinates both equations reduce to suitable singular Hamilton–Jacobi first order equations. After considering the case of monotone data we point out a new peculiar behavior for non-monotone data with a profile of Batman type ( g = min { g 1, g 2 }, g 1 ( r ) increasing, g 2 ( r ) decreasing and g 1 ( r d ) = g 2 ( r d ) for some r d > 0 ). In the parabolic regime, and when the velocity of the convexity part of the level sets is greater than the velocity of the concavity part, we show that the level set { u = g ( r d ) } develops a non-empty interior set for any t > 0 . Nothing similar occurs in the stationary regime. We also present some numerical experiences.