 11125 Timothy Blass, Rafael de la Llave, Enrico Valdinoci
 A comparison principle for a Sobolev gradient semiflow
(429K, pdf)
Sep 2, 11

Abstract ,
Paper (src),
View paper
(auto. generated pdf),
Index
of related papers

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 AubryMather theory for partial
differential equations and pseudodifferential equations by
finding planelike minimizers of the energy functional.
 Files:
11125.src(
11125.comments ,
11125.keywords ,
Blass_Llave_ValdinociCPAA.pdf.mm )