A Comparison Principle for a Sobolev Gradient Semi-flow

We consider gradient descent equations for energy functionals of the type S(u) = 1 2 〈u(x), A(x)u(x)〉L2 + ∫ Ω 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 , β ∈ (0, 1), with a metric that depends on A and a positive number γ > sup |V22|. 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 planelike minimizers of the energy functional.