A comparison principle for a Sobolev gradient semi-flow

We consider gradient descent equations for energy functionals of the type S(u) = 12 hu(x);A(x)u(x)iL2 +R V (x; u) dx, where A is a uniformly elliptic operator of order 2, with smooth coeffcients. 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,β β ε(0; 1), with a metric that depends on A and a positive number γ β sup jV22j. We prove a weak comparison principle for such a gradient ow.