PreprintCC BY 4.0HAL
Complex Monge-Ampère equations with boundary values: a variational approach
Abstract
Let Ω ⋐ C n be a bounded hyperconvex domain and ϕ : Ω → R a maximal plurisubharmonic function in Ω (possibly unbounded). We introduce a new class E 1 ϕ (Ω) of plurisubharmonic functions with boundary values equal to ϕ in the sense of Cegrell and finite ϕ-energy on Ω. Then given a Borel measure µ on Ω, we apply a variational approach to prove that µ is the Monge-Ampère measure of a function u ∈ E 1 ϕ (Ω) if and only if it satisfies the integrability condition
We also establish a quantitative version of this condition extending previous results of Cegrell and others. Using a logarithmic transformation, we obtain analogous results for the real Monge-Ampère equations with (unbounded) boundary data.
Related
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BibTeX
@misc{badiane2026complex,
title = {{Complex Monge-Ampère equations with boundary values: a variational approach}},
author = {Papa Badiane and Chinh H. Lu and Souhaibou Sambou and Ahmed Zeriahi},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13478},
eprint = {2610.13478},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05787570v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complex Monge-Ampère equations with boundary values: a variational approach
AU - Papa Badiane
AU - Chinh H. Lu
AU - Souhaibou Sambou
AU - Ahmed Zeriahi
PY - 2026
DA - 2026/10/08
PB - HAL
DO - 10.48550/arxiv.2610.13478
UR - https://hal.science/hal-05787570v1
AB - Let Ω ⋐ C n be a bounded hyperconvex domain and ϕ : Ω → R a maximal plurisubharmonic function in Ω (possibly unbounded). We introduce a new class E 1 ϕ (Ω) of plurisubharmonic functions with boundary values equal to ϕ in the sense of Cegrell and finite ϕ-energy on Ω. Then given a Borel measure µ on Ω, we apply a variational approach to prove that µ is the Monge-Ampère measure of a function u ∈ E 1 ϕ (Ω) if and only if it satisfies the integrability condition We also establish a quantitative version of this condition extending previous results of Cegrell and others. Using a logarithmic transformation, we obtain analogous results for the real Monge-Ampère equations with (unbounded) boundary data.
ER -