 09148 T. Blass, R. de la Llave, E. Valdinoci
 A Comparison Principle for a Sobolev Gradient SemiFlow
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Aug 26, 09

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Abstract. We consider gradient descent equations for energy functionals of
the type $S(u) = \frac{1}{2}\langle u(x), A(x)u(x) \rangle_{L^2} +
\int_{\Omega} V(x,u) \, dx$, 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 AubryMather theory for partial
differential equations and pseudodifferential equations.
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