[Paper Review] Regularity for solutions of non local parabolic equations

Héctor A. Chang Lara, Gonzalo Dávila|arXiv (Cornell University)|Sep 15, 2011

Advanced Mathematical Physics Problems19 references6 citations

TL;DR

This paper establishes optimal H"{o}lder and $C^{1,eta}$ regularity estimates for solutions of fully nonlinear nonlocal parabolic equations with translation-invariant kernels, using a weak parabolic ABP estimate and iterative oscillation decay. The results remain uniform as the order of the equation approaches the classical local case ($\sigma \to 2$), recovering and extending classical regularity theory in the nonlocal setting.

ABSTRACT

We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We proof Holder regularity in space and time and for translation invariant equations and under different assumptions on the kernels Holder regularity for the spatial derivatives. The proofs rely on a weak parabolic ABP inspired in recent work done by L. Silvestre and the classic ideas of K. Tso and L. Wang. Our results remain uniform as the order of the equation goes to 2 allowing us to understand the non local theory as an extension to the classical one.

Motivation & Objective

• To establish $C^{\alpha}$ regularity in space and time for viscosity solutions of fully nonlinear nonlocal parabolic equations.
• To prove $C^{1,\alpha}$ regularity in space under translation-invariant kernel assumptions.
• To ensure the regularity estimates remain uniform as the order $\sigma \to 2$, recovering classical results from [10].
• To develop a weak parabolic ABP estimate that enables iterative oscillation decay in the nonlocal setting.

Proposed method

• Develops a weak parabolic Alexandroff-Bitsadze-Pucci (ABP) estimate by covering the contact set of $u$ with its convex envelope $\Gamma$ using sets where $u$ does not deviate significantly from $\Gamma$.
• Uses iterative application of a modified version of Lemma 5.1 from [8] to control the upper bound of the convex envelope $\Gamma$.
• Employs an appropriate barrier function to control the lower bound of $\Gamma$, ensuring it does not decrease too rapidly.
• Applies a pointwise oscillation decay estimate inspired by Wang's method in [10], adapted to the nonlocal setting.
• Uses incremental quotients $w^{h,k}$ and cut-off functions $\eta^k$ to localize the problem and control Hölder norms iteratively.
• Establishes uniform bounds on $w^{h,k}$ and $w_{1}^{h,k}$ via viscosity inequalities involving the maximal and minimal operators $\mathcal{M}^{\pm}_{\mathcal{L}_1}$, leading to $C^{1,\alpha}$ regularity.

Experimental results

Research questions

• RQ1 Can $C^{\alpha}$ regularity in space and time be established for viscosity solutions of fully nonlinear nonlocal parabolic equations?
• RQ2 Does $C^{1,\alpha}$ regularity in space hold under translation-invariant kernel assumptions, and is it uniform as $\sigma \to 2$?
• RQ3 Can a weak parabolic ABP estimate be constructed in the nonlocal setting to enable oscillation decay arguments?
• RQ4 How can the classical regularity theory of [10] be extended to the nonlocal case with uniform estimates as $\sigma \to 2$?
• RQ5 What role do the maximal and minimal operators $\mathcal{M}^{\pm}_{\mathcal{L}}$ play in controlling the oscillation of incremental quotients?

Key findings

• The paper proves $C^{\alpha}$ regularity in space and time for solutions of nonlocal parabolic equations via a weak parabolic ABP estimate and oscillation decay.
• Under translation-invariant kernel assumptions, the authors establish $C^{1,\alpha}$ regularity in space, with the estimates uniform as $\sigma \to 2$.
• The weak ABP estimate is constructed by covering the contact set of $u$ with $\Gamma$ using regions where $u$ stays close to $\Gamma$, relying on iterative control of $\Gamma$ from above and below.
• The incremental quotient $w^{h,k}$ satisfies viscosity inequalities involving $\mathcal{M}^{\pm}_{\mathcal{L}_1}$, and uniform bounds on $w^{h,k}$ are propagated iteratively to yield $C^{1,\alpha}$ regularity.
• The method yields uniform regularity estimates as $\sigma \to 2$, allowing recovery of most results from Wang's classical work [10] in the nonlocal setting.
• The final result confirms that $u_x$ is bounded in $B_{1/2} \times [-1/2, 0]$, and higher regularity follows by one more iteration on the Lipschitz quotient.