11-125 Timothy Blass, Rafael de la Llave, Enrico Valdinoci

A comparison principle for a Sobolev gradient semi-flow (429K, pdf) Sep 2, 11

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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.

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