AI 中文总结
本文研究Banach空间中eikonal方程可微解的正则性，证明底层空间几何决定解的C^{1,1}正则性与仿射性，并应用于刻画梯度值域受限的可微函数。
AI 中文摘要
eikonal方程的粘性解不一定具有超越Lipschitz连续性的正则性。相比之下，可微解表现出更强的结构性质。在欧几里得空间中，基于Caffarelli和Crandall（Comm. Partial Differential Equations $\textbf{35}$ (2010), 391--414）的观察，人们发现eikonal方程$\\| Df \\| \equiv 1 \\,\\, \text{in } \mathcal{U} \subset \mathbb{R}^d$的可微解属于$C^{1, 1}_{\mathrm{loc}}(\mathcal U)$类。此外，如果$\mathcal{U} = \mathbb{R}^d$，则$f$必须是仿射的。本手稿在（有限维和无限维）Banach空间的背景下重新审视这些现象。我们的结果表明，底层Banach空间的几何结构如何决定eikonal方程的Gateaux可微解的正则性和仿射性。作为有限维中的一个几何应用，我们刻画了梯度值域被限制在凸体边界上的可微函数。
英文摘要
Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations $\textbf{35}$ (2010), 391--414), one finds that differentiable solutions of the eikonal equation $\| Df \| \equiv 1 \,\, \text{in } \mathcal{U} \subset \mathbb{R}^d$ are of class $C^{1, 1}_{\mathrm{loc}}(\mathcal U)$. Moreover, if $\mathcal{U} = \mathbb{R}^d$, then $f$ must be affine. This manuscript revisits these phenomena in the setting of (finite-- and infinite--dimensional) Banach spaces. Our results show how the geometry of the underlying Banach space determines the regularity and affineness of Gateaux--differentiable solutions of eikonal equations. As a geometric application in finite dimensions, we characterize differentiable functions whose gradient ranges are constrained to boundaries of convex bodies.
