Symmetry and monotonicity of solutions to elliptic and parabolic fractional $p$-equations

TL;DR

This work develops a maximum principle for antisymmetric functions in parabolic fractional $p$-equations and leverages parabolic inequalities to furnish a new proof of symmetry and monotonicity for elliptic fractional $p$-Laplacian problems with gradient terms. By applying a moving planes method to the parabolic setting, the authors then establish symmetry and monotonicity for nonlinear parabolic fractional $p$-equations on both bounded domains and the whole space, including radially symmetric and monotone profiles in the unit ball and decaying solutions in $\,\\mathbb{R}^n$. The key innovations include the damping transformation $ ilde{u}_\lambda=e^{-mt}u_\lambda$ to handle gradient terms and a robust maximum principle for antisymmetric nonlocal operators. The results generalize known linear cases and provide a versatile toolkit for nonlinear nonlocal elliptic and parabolic problems involving the fractional $p$-Laplacian.

Abstract

In this article we first establish the maximum principle of the antisymmetric functions for parabolic fractional $p$-equations. Then we use it and the parabolic inequalities to provide a different proof of symmetry and monotonicity for solutions to elliptic fractional $p$-equations with gradient terms. Finally, base on suitable initial value, by the maximum principle of the antisymmetric functions for parabolic fractional $p$-equations, we attain symmetry and monotonicity of positive solutions in each finite time to nonlinear parabolic fractional $p$-equations on the whole space and bounded domains. We believe that the maximum principle and parabolic inequalities obtained here can be utilized to many elliptic and parabolic problems involving nonlinear nonlocal operators.

Theorems & Definitions (12)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 2.1
• Theorem 2.2
• Remark 2.3