Global Lipschitz Regularity of Degenerate Monge-Ampere Eigenfunctions on Convex Domains
The nonzero convex solutions of the degenerate Monge-Ampere equation det D^2 u = M|u|^p with zero boundary data on a bounded convex domain are globally Lipschitz exactly when p exceeds the sharp threshold n-2, which also yields global W^{2,1} Sobolev bounds on the Hessian. The result is proved by constructing explicit Lipschitz convex subsolutions and applying a comparison principle for degenerate subcritical Monge-Ampere equations; at the borderline p = n-2 the boundary gradient can blow up, showing the threshold is optimal.
GLOBAL LIPSCHITZ AND SOBOLEV ESTIMATES FOR THE MONGE–AMP`ERE EIGENFUNCTIONS OF GENERAL BOUNDED
This paper proves that the Monge-Ampere eigenfunctions of a general bounded convex domain -- the nonzero convex solutions u of det D^2 u = lambda |u|^n with u = 0 on the boundary -- are globally Lips…