On Singular Boundary Value Problems for the Monge–Ampére Operator

Abstract

We consider different types of singular boundary value problems for the Monge᎐Ampere operator. The approach is based on existing regularity theory and a subsolution᎐supersolution method. Nonexistence and uniqueness results are also given. ᮊ

On the existence of solutions to the Monge-Ampère equation with infinite boundary values

“…In the paper [19], the authors show that problems (1.1) and (1.2) have no solution when g ∈ C ∞ (Ω) and f (t) = t γ , 0 < γ ≤ n. The next result extends this nonexistence result to include g ∈ C ∞ (Ω) and nonlinearities f that satisfy the following condition: Proof. Suppose u ∈ C 2 (Ω) is a strictly convex function that satisfies det D 2 u ≤ gf (u) on Ω and the boundary condition (1.2).…”

“…The following comparison lemma is well known [14,19] and will be used repeatedly in subsequent proofs. Since we state it in a slightly different form for our purpose, we have included a short proof for completeness.…”

“…The argument rests on well-known a priori estimates for solutions of (3.2) established in the papers [4,26]. Such arguments have been used in [19] when τ > −∞ and in [14,20,25] for the case τ = −∞. For completeness we will provide the argument when τ > −∞ in our case.…”

“…Let f be a positive and an increasing smooth function on (τ, ∞) for some −∞ ≤ τ < ∞. In this paper we will be concerned with convex solutions of the Monge-Ampère equation This problem was considered in [6,7,19,20], and more recently in [14]. In all these papers, the function g was assumed to be bounded on Ω.…”

“…In all these papers, the function g was assumed to be bounded on Ω. Moreover, [6,7,19] consider the case when f (t) = t γ or when f (t) = exp(kt) for some positive γ > n and k. In [14], the authors consider a much more general case with the right-hand side of (1.1) depending on the gradient Du as well.…”