A Bernstein theorem for affine maximal type hypersurfaces under decaying convexity. (October 2019) - Du, Shi-Zhong
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.