Timothy Blass, Rafael de la Llave, Enrico Valdinoci
A comparison principle for a Sobolev gradient semi-flow
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ABSTRACT.  We consider gradient descent equations for energy functionals of 
the type $S(u) = rac{1}{2}\langle u(x), A(x)u(x) 
angle_{L^2} + 
\int_{\Omega} V(x,u) \, dx$, 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 
$H^{eta}$, $eta \in (0,1)$, 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, 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.