Perturbation and numerical methods for computing the minimal average energy
Timothy Blass Rafael de la Llave
Networks & Heterogeneous Media 2011, 6(2): 241-255 doi: 10.3934/nhm.2011.6.241
We investigate the differentiability of minimal average energy associated to the functionals $S_\epsilon (u) = \int_{\mathbb{R}^d} \frac{1}{2}|\nabla u|^2 + \epsilon V(x,u)\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter $\epsilon$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.
keywords: Lindstedt series Minimal average energy Sobolev gradient descent Plane-like minimizers Cell problem quasiperiodic solutions of PDE.
A comparison principle for a Sobolev gradient semi-flow
Timothy Blass Rafael De La Llave Enrico Valdinoci
Communications on Pure & Applied Analysis 2011, 10(1): 69-91 doi: 10.3934/cpaa.2011.10.69
We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} < u(x), A(x)u(x)>_{L^2} + \int_{\Omega} V(x,u) dx$, 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$|V_{2 2}|$. 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.
keywords: Comparison principle fractional powers of elliptic operators. Sobolev gradient semigroups of linear operators

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