---
tag: X0Q69ZS09T
kind: concept
name: Convex integrand
msc: [49J45, 26B25]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_function
---

## Summary

A convex integrand is a Lagrangian $L(x, u, \xi)$ of an integral functional $\int_\Omega L(x, u, \nabla u) \, dx$ that is a [[convex function|07GR3TNW4H]] of the gradient variable $\xi$.
Convexity in $\xi$ is the hypothesis that makes such a functional weakly lower semicontinuous on Sobolev spaces, for scalar functions $u$.
It is therefore the basic structural assumption of existence theorems such as Tonelli's.
For vector-valued $u$ it can be weakened to quasiconvexity.

## Explanations

### L1 human

The integrand describes the cost of each local slope of a function.
If this cost is convex in the slope, mixing two slopes in fine oscillations never costs less than using their average slope.
So oscillating functions cannot undercut smooth ones, and a minimiser can be found as a limit.
The Dirichlet energy, with $L = \frac{1}{2} \abs{\xi}^2$, has a convex integrand.
The double-well cost $(\xi^2 - 1)^2$ is not convex, and indeed fine zigzags with slopes $\pm 1$ beat the flat function.