Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation

Abstract

Given a strictly convex, smooth, and bounded domain Ω in R n we establish the existence of a negative convex solution in C ∞ (Ω) ∩ C(Ω) with zero boundary value to the singular Monge-Ampère equation det(D 2 u) = p(x)g(−u). An associated Dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem. Estimates of solutions will also be given and regularity of solutions will be deduced from the estimates.

Optimal global asymptotic behavior of the solution to a singular monge-ampère equation

“…In [18], Mohammed extended the results in [3] and [12] for more general g and b, and obtained the following results. Let b satisfy (B 1 ) and g be a positive nonincreasing smooth function in (0, ∞).…”

“…[18], Theorem 2.1). Let b ∈ C ∞ (Ω) be positive on Ω and g be a positive nonincreasing smooth function in (0, ∞).…”

“…There are many papers which have been dedicated to resolving the global and boundary regularity and asymptotic behavior issues to the problem, see, for instance, [3,4], [7]- [12], [14,15,18,19,22,26] and the references therein.…”

Strictly convex solutions for singular Monge–Ampère equations with nonlinear gradient terms: existence and boundary asymptotic behavior

“…where b ¼ nþ1 nþc and dðxÞ ¼ distðx; oXÞ. Recently, Mohammed [26] established the existence and the global estimates of solutions of the Monge-Ampère problem:…”

Positive Solutions of Two-Point Boundary Value Problems for Monge-Ampère Equations

“…For the case that ( ) is singular at = 0, there are fewer results for BVP (1). But some interesting results are presented for BVP (2) in [9][10][11][12] where ( ) is singular at = 0. We also refer to [7,8,13,14] and references therein for further discussions regarding solutions of the Monge-Ampère equations.…”