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.