# On problems in the calculus of variations in increasingly elongated domains

## TL;DR

This paper investigates the asymptotic behavior of calculus of variations minimizers in elongated domains, showing convergence to minimizers of a related functional in the fixed directions as the domain size increases.

## Contribution

It introduces a framework for analyzing the asymptotic limits of variational problems in elongated domains with data depending only on certain coordinates.

## Key findings

- Minimizers converge to a limit functional in fixed directions
- As domain elongates, the problem simplifies to a lower-dimensional variational problem
- Results apply to various configurations of elongated domains

## Abstract

We consider minimization problems in the calculus of variations set in a sequence of domains the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. We study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions.