[Paper Review] Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional $p$-Laplacians

Pengyan Wang, Wen‐Xiong Chen|arXiv (Cornell University)|Oct 2, 2020

Nonlinear Partial Differential Equations30 references4 citations

TL;DR

This paper establishes parabolic versions of Hopf's lemma for fractional $p$-Laplacian equations, proving that solutions vanish with a precise fractional order $s$ near the boundary and that antisymmetric solutions satisfy an asymptotic Hopf-type boundary behavior. These results enable the method of moving planes to prove symmetry and monotonicity of solutions to nonlocal parabolic equations.

ABSTRACT

In this paper, we first establish Hopf's lemmas for parabolic fractional equations and parabolic fractional $p$-equations. Then we derive an asymptotic Hopf's lemma for antisymmetric solutions to parabolic fractional equations. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.

Motivation & Objective

• To extend classical Hopf's lemma to parabolic equations involving nonlocal fractional $p$-Laplacian operators.
• To address the lack of boundary behavior results for parabolic nonlocal equations, especially regarding the normal derivative at the boundary.
• To derive an asymptotic Hopf's lemma for antisymmetric solutions, crucial for the method of moving planes in the parabolic setting.
• To provide tools for proving qualitative properties such as symmetry, monotonicity, and non-existence of solutions in nonlocal parabolic PDEs.

Proposed method

• Establish a parabolic Hopf's lemma for solutions to $ \frac{\partial u}{\partial t} + (-\Delta)^s_p u = f(t,u) $ in a bounded domain $ \Omega \times (0,T] $ with zero Dirichlet boundary conditions.
• Prove that $ u(x,t_0) \geq c_0 d^s(x) $ near the boundary, where $ d(x) = \text{dist}(x,\partial\Omega) $, implying the fractional normal derivative $ \frac{\partial u}{\partial \nu^s} < 0 $ at the boundary.
• Introduce an asymptotic Hopf's lemma for antisymmetric functions $ w_\lambda(x,t) = u(x^\lambda,t) - u(x,t) $, showing $ \frac{\partial \psi_\lambda}{\partial \nu} < 0 $ on the boundary of the half-space $ \Sigma_\lambda $.
• Use the $ \omega $-limit set $ \omega(u) $ to analyze long-time behavior and apply the asymptotic strong maximum principle to derive contradiction in symmetry proofs.
• Apply the results in the method of moving planes by verifying the initial position of the plane using the boundary behavior and then proving symmetry via contradiction.
• Leverage regularity and compactness in $ C_0(\mathbb{R}^n) $ to pass limits and ensure the existence of symmetric $ \omega $-limit profiles.

Experimental results

Research questions

• RQ1 Can a parabolic version of Hopf's lemma be established for fractional $ p $-Laplacian equations?
• RQ2 What is the precise boundary behavior of solutions to parabolic fractional $ p $-Laplacian equations near the domain boundary?
• RQ3 How can the method of moving planes be adapted for nonlocal parabolic equations using boundary estimates?
• RQ4 What asymptotic boundary behavior do antisymmetric solutions exhibit in the context of the method of moving planes?
• RQ5 Can the asymptotic Hopf's lemma be used to prove symmetry and monotonicity of solutions to parabolic fractional $ p $-Laplacian equations?

Key findings

• For parabolic fractional $ p $-Laplacian equations with zero Dirichlet data, solutions satisfy $ u(x,t_0) \geq c_0 d^s(x) $ near the boundary, implying the fractional normal derivative is negative.
• The fractional normal derivative $ \frac{\partial u}{\partial \nu^s} < 0 $ at the boundary, which is essential for initiating the method of moving planes.
• An asymptotic Hopf's lemma holds for antisymmetric functions: if $ \psi_\lambda \geq 0 $ and not identically zero, then $ \frac{\partial \psi_\lambda}{\partial \nu} < 0 $ on $ \partial \Sigma_\lambda $.
• The asymptotic Hopf's lemma enables the second step of the method of moving planes by ensuring that the solution cannot attain a minimum on the boundary unless it is identically zero.
• The contradiction argument in the moving planes method is closed by showing that a negative minimum with zero gradient on the boundary contradicts the strict negativity of the normal derivative.
• The $ \omega $-limit set $ \omega(u) $ is compact in $ C_0(\mathbb{R}^n) $, ensuring the existence of symmetric limit profiles under mild boundedness assumptions.