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Sharp Small-Height Estimates for Subcritical Special Lagrangian Equations in Dimension Three
Abstract
We prove an interior Lipschitz estimate with a sharp height threshold for continuous viscosity solutions of the three-dimensional special Lagrangian equation with subcritical phase |Θ|<π/2. If oscBRu≤ MR2, 0≤ M<secΘ, then LipBρRu≤ CR for some ρ>0 and C>0 depending only on |Θ| and M. The threshold is optimal: at M=secΘ, there are entire real-analytic solutions whose Lipschitz seminorms are unbounded on every fixed interior ball. The proof uses a two-point maximum principle and algebraic identities specific to dimension three.
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BibTeX
@misc{li2026sharp,
title = {{Sharp Small-Height Estimates for Subcritical Special Lagrangian Equations in Dimension Three}},
author = {Caiyan Li and Wei Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01986},
eprint = {2610.01986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01986},
}
RIS
TY - UNPB
TI - Sharp Small-Height Estimates for Subcritical Special Lagrangian Equations in Dimension Three
AU - Caiyan Li
AU - Wei Wei
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01986
UR - https://arxiv.org/abs/2610.01986
AB - We prove an interior Lipschitz estimate with a sharp height threshold for continuous viscosity solutions of the three-dimensional special Lagrangian equation with subcritical phase |Θ|<π/2. If osc_B_Ru≤ MR^2, 0≤ M<secΘ, then Lip_B_ρRu≤ CR for some ρ>0 and C>0 depending only on |Θ| and M. The threshold is optimal: at M=secΘ, there are entire real-analytic solutions whose Lipschitz seminorms are unbounded on every fixed interior ball. The proof uses a two-point maximum principle and algebraic identities specific to dimension three.
ER -