[Paper Review] On Schauder estimates for a class of nonlocal fully nonlinear parabolic equations

Hongjie Dong, Hong Zhang|arXiv (Cornell University)|Apr 1, 2016

Nonlinear Partial Differential Equations19 references3 citations

TL;DR

This paper establishes Schauder estimates for concave fully nonlinear nonlocal parabolic equations with rough, non-symmetric kernels of order $\sigma \in (0,2)$, proving that solutions are $C^{1+\alpha/\sigma,\sigma+\alpha}$ regular under Hölder continuity assumptions on the data and kernels. The key contribution is extending Evans-Krylov-type estimates to non-symmetric, non-smooth kernels, with uniform bounds as $\sigma \to 2$, and deriving new results for nonlocal elliptic equations as a corollary.

ABSTRACT

We obtain Schauder estimates for a class of concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. As a application, we prove that the solution to a translation invariant equation with merely bounded data is $C^σ$ in $x$ variable and $Λ^1$ in $t$ variable, where $Λ^1$ is the Zygmund space.

Motivation & Objective

• To extend Schauder estimates to fully nonlinear nonlocal parabolic equations with non-symmetric and rough kernels, which are not covered by prior results assuming symmetry or smoothness.
• To prove that solutions to translation-invariant equations with merely bounded data are $C^{\sigma}$ in space and $\Lambda^1$ in time, even without regularity assumptions on the kernel.
• To derive new Schauder estimates for nonlocal elliptic equations with rough and non-symmetric kernels, which are novel contributions in this setting.
• To establish uniform bounds in the limit $\sigma \to 2$, recovering classical Schauder estimates in the local case.

Proposed method

• The authors use a blow-up analysis and a Liouville-type theorem to prove regularity estimates without assuming symmetry or smoothness of the kernel.
• They introduce a modified version of the classical interpolation inequality to control Hölder seminorms of solutions in space and time variables.
• A key technical step involves constructing a cutoff function and using a nonlocal version of the $\Lambda^1$ time regularity to control time oscillations.
• The proof relies on scaling arguments and localization techniques to derive a priori estimates in dyadic cylinders, leveraging the concavity of the equation.
• They establish a recursive decay estimate for the oscillation of solutions in annular regions, which leads to Hölder regularity in space and time.
• The method applies to both parabolic and elliptic equations by reducing the elliptic case to the parabolic one via a time-averaging argument.

Experimental results

Research questions

• RQ1 Can Schauder estimates be established for fully nonlinear nonlocal parabolic equations with non-symmetric and rough kernels, without assuming smoothness or symmetry of the kernel?
• RQ2 What is the optimal regularity of solutions to translation-invariant nonlocal parabolic equations with bounded data and rough kernels?
• RQ3 How do the regularity estimates behave in the limit $\sigma \to 2$, and do they recover classical Schauder estimates?
• RQ4 Can the Evans-Krylov theorem be extended to non-symmetric kernels without requiring $C^2$ regularity in the kernel?
• RQ5 What is the role of the Zygmund space $\Lambda^1$ in characterizing time regularity for nonlocal equations?

Key findings

• The solution $u$ to the equation $u_t = \inf_{a \in \mathcal{A}}(L_a u + f_a)$ satisfies the a priori estimate $[u]_{1+\alpha/\sigma, \sigma+\alpha; Q_{1/2}} \leq C \|u\|_{\alpha/\sigma, \alpha; (-1,0) \times \mathbb{R}^d} + C C_0$, where $C_0 = \sup_a [f_a]_{\alpha/\sigma, \alpha; Q_1}$.
• The constant $C$ in the estimate depends only on $d, \lambda, \Lambda, \alpha, A, \sigma$, and remains uniformly bounded as $\sigma \to 2$, ensuring continuity with classical theory.
• Solutions to translation-invariant equations with bounded data are shown to be $C^\sigma$ in space and $\Lambda^1$ in time, even when the kernel is non-symmetric and rough.
• The authors prove that $C^{1+\alpha/\sigma, \sigma+\alpha}$ regularity holds for any $\alpha \in (0, \hat{\alpha})$ with $\hat{\alpha} \in (0,1)$ depending on $d, \sigma, \lambda, \Lambda$, under Hölder continuity assumptions on $f_a$ and $K_a$.
• The results for nonlocal parabolic equations imply new Schauder estimates for nonlocal elliptic equations with rough and non-symmetric kernels, which were previously unknown.
• The proof technique, based on blow-up analysis and Liouville-type theorems, allows handling non-symmetric kernels without requiring the cancellation condition $\int_{S_r} y K_a(t,x,y) \, ds = 0$ for $\sigma = 1$, though this condition is assumed in the case $\sigma = 1$.