超线性哈密顿量的非局部Hamilton-Jacobi方程的改进正则性
Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
- Université Paris Cité, CNRS, Sorbonne Université（巴黎西岱大学、法国国家科学研究中心、索邦大学）
- O. Mayer Mathematics Institute, Romanian Academy（罗马尼亚科学院 O. Mayer 数学研究所）
- Univ Rennes, INSA Rennes, CNRS（雷恩大学、雷恩国立应用科学学院、法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
研究一类由x依赖积分微分算子和强制超线性哈密顿量驱动的非局部Hamilton-Jacobi方程的粘性解正则性，在一般结构假设下建立Hölder正则性，并在弱椭圆性条件下证明全局Lipschitz连续性，通过反例说明假设的尖锐性。
中文摘要 AI 辅助
我们研究由依赖于x的积分微分算子和强制超线性哈密顿量驱动的一类非局部Hamilton-Jacobi方程的粘性解的正则性。我们首先在底层Lévy测度的一般结构和连续性假设下，建立了有界粘性解的Hölder正则性，而不对非局部算子施加任何椭圆性条件。Hölder指数根据算子的阶σ∈(0,2)和哈密顿量的增长指数m>1显式给出。特别地，我们的方法适用于任意超线性哈密顿量，包括1<m<σ<2的精细情形，并在σ∈(1,2)时产生改进的正则性指数。进一步假设非局部算子满足弱椭圆性条件，我们证明粘性解是全局Lipschitz连续的。证明将强制哈密顿量提供的Hölder正则性与非局部扩散的正则化效应通过Ishii-Lions论证相结合，使我们能够处理具有任意超线性增长的哈密顿量。最后，我们提供一个反例，表明在没有椭圆性的情况下，如果Lévy测度的空间依赖性仅为Hölder连续，则Lipschitz正则性可能失效，从而说明我们连续性假设的尖锐性。
英文摘要
We investigate the regularity of viscosity solutions to a class of nonlocal Hamilton-Jacobi equations driven by x-dependent integro-differential operators and coercive superlinear Hamiltonians. We first establish H{ö}lder regularity for bounded viscosity solutions under general structural and continuity assumptions on the underlying L{é}vy measures, without imposing any ellipticity condition on the nonlocal operator. The H{ö}lder exponent is given explicitly in terms of the order $σ$ $\in$ (0, 2) of the operator and the growth exponent m > 1 of the Hamiltonian. In particular, our approach applies to arbitrary superlinear Hamiltonians, including the delicate regime 1 < m < $σ$ < 2, and yields an improved regularity exponent when $σ$ $\in$ (1, 2). Assuming in addition a weak ellipticity condition on the nonlocal operator, we prove that viscosity solutions are globally Lipschitz continuous. The proof combines the H{ö}lder regularity supplied by the coercive Hamiltonian with the regularizing effect of the nonlocal diffusion through an Ishii-Lions argument, allowing us to treat Hamiltonians with arbitrary superlinear growth. Finally, we provide a counterexample showing that, in the absence of ellipticity, Lipschitz regularity may fail if the spatial dependence of the L{é}vy measures is merely H{ö}lder continuous, thereby illustrating the sharpness of our continuity assumptions.
