超二次哈密顿-雅可比方程的极大正则性：Lions猜想的端点情形

中文摘要 AI 辅助

P.-L. Lions的著名猜想涉及粘性哈密顿-雅可比方程的极大正则性，本文研究其端点情形。我们考虑d≥2、γ>2、f属于L^{q_c}(T^d)（其中q_c=d(γ-1)/γ）时，环面T^d上方程-Δu + |Du|^γ = f的归一化强解。在该临界指数下，主要困难是临界标度下可能出现的集中现象。假设源项构成L^{q_c}的一致等可积子集，我们排除这种集中并证明强解的极大L^{q_c}正则性。证明结合了两阶段爆破论证与Liouville刚性定理：临界局部能量估计排除了第一爆破尺度下梯度能量在L^{γ q_c}中的集中，而小漂移正则性将爆破序列的弱紧性升级为强局部紧性。

英文摘要

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 $\mathbb T^d$, where $d\geq 2$, $γ>2$, and $f\in L^{q_c}(\mathbb T^d)$ 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.

发表机构

• University of Washington（华盛顿大学）

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