On the Hölder continuous subsolution problem for the complex Monge–Ampère equation

Abstract

We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge-Ampère equation admits Hölder continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge-Ampère total mass, we obtain an affirmative answer to a question of Zeriahi et al. (Complex Var. Elliptic Equ. 61(7):902-930, 2016).

A Remark on the Continuous Subsolution Problem for the Complex Monge-Ampère Equation

“…In particular, if μ has density with respect to the Lebesgue measure in L p , p > 1 then this bound is satisfied [9]. By the recent results in [12,13] if μ is bounded by the Monge-Ampère measure of a Hölder continuous plurisubharmonic function ϕ μ ≤ (dd c ϕ) n in Ω, then (1.2) holds for a specific h, and consequently, the Dirichlet problem (1.1) is solvable with Hölder continuous solution. The main result in this paper says that we can considerably weaken the assumption on ϕ and still get a continuous solution of the equation.…”

“…Since μ ∈ H(α, Ω), according to [9] and [10, Theorem 5.9] we can solve the Dirichlet problem (1.1) to obtain a unique continuous solution u. Define for δ > 0 small Thanks to the arguments in [12,Lemma 2.11] it is easy to see that there exists δ 0 > 0 such that u δ (z) ≤ u(z) + (δ; ψ, ∂Ω) (4.1) for every z ∈ ∂Ω δ and 0 < δ < δ 0 . Here, we used the result of Bedford and Taylor [3, Theorem 6.2] (with minor modifications) to extend ψ plurisubharmonically onto Ω so that its modulus of continuity onΩ is controlled by the one on the boundary.…”

Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

“…Then, we can assume μ is compactly supported in U and μ ≤ (dd c ϕ) n for some Hölder continuous plurisubharmonic function ϕ in . By [12,Corollary 1.2] (see also [33,Lemma 2.7, Proposition 2.9]) we get that μ is moderate andμ is Hölder continuous on E 0 ( ). Thus, it is also Hölder continuous on S.…”

“…By [33, Lemma 2.15, Corollary 2.14] it follows thatμ is Hölder continuous on E 0 ( ), then so is the functional of f dμ. Finally, by [33,Propositon 2.9] we have that f dμ is moderate.…”

“…There are plenty of examples of measures which are locally Hölder continuous on S 0 ( ) (see [33,34]). We give below a sufficient condition.…”