Open AccessNeumann problem for Monge-Ampere type equations revisited.
Author(s) -
Feida Jiang,
Neil S. Trudinger
Publication year - 2021
Publication title -
new zealand journal of mathematics
Language(s) - English
Resource type - Journals
eISSN - 1179-4984
pISSN - 1171-6096
DOI - 10.53733/176
Subject(s) - mathematics , bounded function , convexity , elliptic operator , domain (mathematical analysis) , degenerate energy levels , mathematical analysis , derivative (finance) , neumann boundary condition , euclidean space , directional derivative , boundary value problem , boundary (topology) , type (biology) , physics , ecology , quantum mechanics , financial economics , economics , biology
This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.