On the Dirichlet problem for degenerate Monge–Ampère type equations

On Neumann problem for the degenerate Monge–Ampère type equations

“…Using the Pogorelov estimates independent of the lower bound B, the interior regularity was established by Błocki [5] and the authors [8]. When A ≡ 0, Andriyanova [3] proved the second order derivative estimates under the A3w condition with the right-hand term B(x, Du) = q(x)ξ (x, Du), where q…”

“…For the Dirichlet boundary value problem, the regularity for degenerate Monge-Ampère equation has been extensively studied, see [3][4][5][6][7][8]. When A ≡ 0, equation (1) reduces to the classical Monge-Ampère equation.…”

“…Theorem 1.1 Let be a bounded C 3,1 domain in R n and u ∈ C 4 ( ) ∩ C 1,1 (¯ ) be an elliptic solution of Neumann (1)- (2). Assume that ϕ ∈ C 2,1 (∂ ), A ∈ C 2 (¯ × R n , R n×n ) satisfies the A3w condition and is uniformly A-convex with respect to u.…”

“…For the boundary estimates, we construct a suitable barrier function which can be used to control the degenerate term, originating from the differentiation of equation (1). In this part, we mix the techniques of dealing with the Neumann problems for nondegenerate Monge-Ampère type equations in [14,15] and the Dirichlet problems for degenerate Monge-Ampère type equations in [3,[5][6][7]. In Sect.…”

$C^{1, 1}$ regularity for solutions to the degenerate $L_p$ Dual Minkowski problem

“…In this case, we encounter with certain degenerate Monge-Ampère type equation. Regularity of solutions to degenerate Monge-Ampère type equations has been investigated in [2,24,25,26,6,11,13,16,41,42,43,27,30] and the references therein. The global C 1,1 regularity of degenerate Monge-Ampère type equations has been obtained in [13] and one cannot expect regularity higher than C 1,1 in general [45].…”