---
tag: 672SS33S5H
kind: definition
name: Convex integrand
msc: [49J45, 26B25]
defines: X0Q69ZS09T
uses: [JRQZ803Q5Y, 07GR3TNW4H]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_function
---

## Definition

Let $\Omega \subseteq \R^n$ be open and let $m \ge 1$.
Let $L \colon \Omega \times \R^m \times \R^{m \times n} \to \R$, written $L(x, u, \xi)$, be the integrand of an [[integral functional|JRQZ803Q5Y]] $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$.
The integrand $L$ is *convex* (in the gradient variable) if for every $x \in \Omega$ and $u \in \R^m$ the map $\xi \mapsto L(x, u, \xi)$ is a [[convex function|07GR3TNW4H]] on $\R^{m \times n}$.

## Remarks

Example: $L(x, u, \xi) = \frac{1}{2} \abs{\xi}^2 - f(x) u$ is convex in $\xi$ for any function $f$.
Indeed $\xi \mapsto \frac{1}{2} \abs{\xi}^2$ is convex, and the term $-f(x) u$ does not depend on $\xi$.

Non-example: $L(\xi) = (\xi^2 - 1)^2$ with $n = m = 1$ is not convex.
Here $L(-1) = L(1) = 0$, but $L(0) = 1 > \frac{1}{2} L(-1) + \frac{1}{2} L(1)$.

A $C^2$ integrand is convex in $\xi$ exactly when its Hessian $\big(\partial^2 L / \partial \xi^i_\alpha \partial \xi^j_\beta\big)$ is positive semidefinite everywhere.
Some authors use the term for integrands that are jointly convex in $(u, \xi)$, which is a stronger condition.
For $m \ge 2$ and $n \ge 2$, convexity is stronger than polyconvexity, quasiconvexity and rank-one convexity.