  We consider gradient descent equations for energy functionals of the type
S(u) = 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_{22}|. 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.
  We 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
