[Paper Review] A doubly nonlinear evolution for ground states of the p-Laplacian

Ryan Hynd, Erik Lindgren|arXiv (Cornell University)|Apr 20, 2014

Nonlinear Partial Differential Equations19 references3 citations

TL;DR

This paper studies the doubly nonlinear PDE $|v_t|^{p-2}v_t = \Delta_p v$ to analyze ground states of the $p$-Laplacian. It proves that the Rayleigh quotient is nonincreasing in time and that $e^{\mu_p t}v(x,t)$ converges to a $p$-ground state as $t \to \infty$, with a limiting equation as $p \to \infty$ offering a novel approximation method for infinity Laplacian ground states.

ABSTRACT

We study the initial value problem related to the PDE $|v_t|^{p-2}v_t=\Delta_p v$. A special property of this equation is that the Rayleigh quotient associated with ground states of the $p$-Laplacian is nonincreasing in time along solutions. Moreover, for each $p\in (1,\infty)$, there is a positive number $\mu_p$ for which $e^{\mu_p t}v(x,t)$ converges to a $p$-ground state as $t$ tends to infinity. An interesting limiting equation also arises when $p$ tends to infinity, which suggests a new way to approximate ground states of the infinity Laplacian.

Motivation & Objective

• To understand the long-time behavior of solutions to the doubly nonlinear PDE $|v_t|^{p-2}v_t = \Delta_p v$.
• To establish that the Rayleigh quotient associated with $p$-Laplacian ground states is nonincreasing along solutions.
• To prove convergence of $e^{\mu_p t}v(x,t)$ to a $p$-ground state as $t \to \infty$ for each $p \in (1,\infty)$.
• To explore the limiting behavior of the equation as $p \to \infty$, yielding a new approach to approximate infinity Laplacian ground states.

Proposed method

• Analyzes the initial value problem for the PDE $|v_t|^{p-2}v_t = \Delta_p v$ using energy methods and comparison principles.
• Uses the Rayleigh quotient associated with the $p$-Laplacian to track monotonicity in time.
• Establishes the existence of a positive constant $\mu_p$ such that $e^{\mu_p t}v(x,t)$ converges to a $p$-ground state as $t \to \infty$.
• Applies asymptotic analysis to the PDE as $p \to \infty$, deriving a limiting equation that characterizes infinity Laplacian ground states.
• Employs variational and spectral techniques to analyze the convergence and stability of ground states.
• Relies on the structure of the $p$-Laplacian and its eigenvalue problem to derive the exponential convergence rate.

Experimental results

Research questions

• RQ1 How does the Rayleigh quotient behave over time for solutions of the PDE $|v_t|^{p-2}v_t = \Delta_p v$?
• RQ2 Does the solution $v(x,t)$ of the PDE converge to a $p$-ground state under exponential scaling as $t \to \infty$?
• RQ3 What is the asymptotic behavior of the PDE as $p \to \infty$, and how does it relate to the infinity Laplacian?
• RQ4 Can the limiting equation as $p \to \infty$ be used to construct approximations of infinity Laplacian ground states?
• RQ5 What is the role of the constant $\mu_p$ in the exponential convergence of $e^{\mu_p t}v(x,t)$ to a $p$-ground state?

Key findings

• The Rayleigh quotient associated with $p$-Laplacian ground states is nonincreasing in time along solutions of the PDE $|v_t|^{p-2}v_t = \Delta_p v$.
• For each $p \in (1,\infty)$, there exists a positive constant $\mu_p$ such that $e^{\mu_p t}v(x,t)$ converges to a $p$-ground state as $t \to \infty$.
• The convergence of $e^{\mu_p t}v(x,t)$ to a $p$-ground state occurs exponentially fast with rate $\mu_p$.
• As $p \to \infty$, the PDE gives rise to a limiting equation that provides a new method for approximating ground states of the infinity Laplacian.
• The limiting equation as $p \to \infty$ suggests a variational and dynamical approach to constructing infinity Laplacian ground states.
• The structure of the PDE ensures that the long-time dynamics are governed by the first eigenfunction of the $p$-Laplacian, with explicit exponential scaling.