A Bernstein theorem for affine maximal type hypersurfaces under decaying convexity. (October 2019)

• Record Type: Journal Article
• Authors: Du, Shi-Zhong
• Is Part Of: Nonlinear analysis. Volume 187(2019)
• Page Start: 170
• Page End: 179
• Publication Date: 2019-10
• Subjects: 53A15 -- 53A10 -- 35J60; Affine maximal hypersurfaces -- Bernstein theorem; 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.2019.04.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we will study a fourth order fully nonlinear PDE u i j D i j w = 0, w ≡ [ det D 2 u ] − θ, θ > 0 of affine maximal type. By a formula of quasi-norm of third derivatives derived from pure PDE tricks, we will prove that any entire convex solution must be a quadratic function, provided θ ∈ ( 0, ( 1 2 − N − 1 2 ( N + 8 ) ) + ) ⋃ ( 1 2 + N − 1 2 ( N + 8 ), + ∞ ) and the minimal convex constant λ ( R ) decays slower than R − α ( θ, N ) for some positive constant α ( θ, N ) given explicitly below.