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### Open Access Journals

NHM

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.

CPAA

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.

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.

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