09-148 T. Blass, R. de la Llave, E. Valdinoci

A Comparison Principle for a Sobolev Gradient Semi-Flow (527K, Ps) 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 Aubry-Mather theory for partial differential equations and pseudo-differential equations.

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