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Small-energy regularity for stationary weak solutions of the Liouville equation
Abstract
We establish small-energy regularity for stationary weak solutions of the Liouville equation $ -Δu=e^u $ in every dimension. It solves a problem left open in Da Lio [Commun. Partial Differ. Equations, 2008] and posed explicitly again by Da Lio and Hyder in [Commun. Contemp. Math. 2025]. The proof relies on the analysis of concentration phenomena for almost stationary fields with structure constraints.
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BibTeX
@misc{fu2026smallenergy,
title = {{Small-energy regularity for stationary weak solutions of the Liouville equation}},
author = {Haotong Fu and Kelei Wang and Wei Wang and Ke Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26026},
eprint = {2609.26026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26026},
}
RIS
TY - UNPB
TI - Small-energy regularity for stationary weak solutions of the Liouville equation
AU - Haotong Fu
AU - Kelei Wang
AU - Wei Wang
AU - Ke Wu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26026
UR - https://arxiv.org/abs/2609.26026
AB - We establish small-energy regularity for stationary weak solutions of the Liouville equation $ -Δu=e^u $ in every dimension. It solves a problem left open in Da Lio [Commun. Partial Differ. Equations, 2008] and posed explicitly again by Da Lio and Hyder in [Commun. Contemp. Math. 2025]. The proof relies on the analysis of concentration phenomena for almost stationary fields with structure constraints.
ER -