Endpoint Maximal Regularity for Superquadratic Hamilton-Jacobi Equations and Applications to Ergodic Mean-field Games Systems
Abstract
A celebrated conjecture of P.-L. Lions concerns maximal regularity for viscous Hamilton--Jacobi equations. In this paper, we study the endpoint case. We consider normalized strong solutions of −Δu+|Du|γ=f in 𝕋d, where d≥ 2, γ>2, and f∈ Lqc(𝕋d) with qc=d(γ−1)/γ. At this critical exponent, the main difficulty is possible concentration under the critical scaling. Assuming that the source terms form a uniformly equi-integrable subset of Lqc, we rule out this concentration and prove maximal Lqc regularity for strong solutions. The proof combines a two-stage blow-up argument with a Liouville rigidity theorem. More precisely, the second blow-up yields a uniform local Lγqc-bound for the gradients, while the small drift arising from the first blow-up upgrades weak convergence to strong local compactness, ultimately leading to a contradiction with Liouville rigidity. Finally, we apply the maximal regularity theory for Hamilton--Jacobi equations at the endpoint case to establish the existence of ergodic solutions to defocusing second-order mean-field games systems with critical coupling exponents.
Related
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BibTeX
@misc{kong2026endpoint,
title = {{Endpoint Maximal Regularity for Superquadratic Hamilton-Jacobi Equations and Applications to Ergodic Mean-field Games Systems}},
author = {Fanze Kong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25066},
eprint = {2608.25066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25066},
}
RIS
TY - UNPB
TI - Endpoint Maximal Regularity for Superquadratic Hamilton-Jacobi Equations and Applications to Ergodic Mean-field Games Systems
AU - Fanze Kong
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.25066
UR - https://arxiv.org/abs/2608.25066
AB - A celebrated conjecture of P.-L. Lions concerns maximal regularity for viscous Hamilton--Jacobi equations. In this paper, we study the endpoint case. We consider normalized strong solutions of −Δu+|Du|^γ=f in 𝕋^d, where d≥ 2, γ>2, and f∈ L^q_c(𝕋^d) with q_c=d(γ−1)/γ. At this critical exponent, the main difficulty is possible concentration under the critical scaling. Assuming that the source terms form a uniformly equi-integrable subset of L^q_c, we rule out this concentration and prove maximal L^q_c regularity for strong solutions. The proof combines a two-stage blow-up argument with a Liouville rigidity theorem. More precisely, the second blow-up yields a uniform local L^γq_c-bound for the gradients, while the small drift arising from the first blow-up upgrades weak convergence to strong local compactness, ultimately leading to a contradiction with Liouville rigidity. Finally, we apply the maximal regularity theory for Hamilton--Jacobi equations at the endpoint case to establish the existence of ergodic solutions to defocusing second-order mean-field games systems with critical coupling exponents.
ER -