A comparison principle for a Sobolev gradient semi-flow

Timothy Blass; Rafael De La Llave; Enrico Valdinoci

doi:10.3934/cpaa.2011.10.69

Article Contents

2011, Volume 10, Issue 1: 69-91. Doi: 10.3934/cpaa.2011.10.69

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A comparison principle for a Sobolev gradient semi-flow

1.
Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257
2.
Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712
3.
Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano

Received: January 2010

Revised: June 2010

Published: January 2011

Abstract / Introduction Related Papers Cited by

Abstract

We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} < u(x), A(x)u(x)>_{L^2} + \int_{\Omega} V(x,u) dx$, where $A$ is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration.
We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow.
We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.

Keywords:

Comparison principle,
Sobolev gradient,
semigroups of linear operators,
fractional powers of elliptic operators.

Mathematics Subject Classification: Primary: 35B50, 46N20; Secondary: 35J20.

Citation:

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