## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

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.

## Year of publication

## Related Authors

## Related Keywords

[Back to Top]