---
tag: 8D455SA1D4
kind: definition
name: Weierstrass excess function
msc: [49K05]
defines: R8TWRPSPNV
uses: [JRQZ803Q5Y, WMPCR723MJ, 05AXG3QRKR]
sources:
  - url: https://en.wikipedia.org/wiki/Weierstrass%E2%80%93Erdmann_condition
---

## Definition

Let $a < b$, let $n \ge 1$ and let $L \colon [a, b] \times \R^n \times \R^n \to \R$, written $L(x, y, p)$, be of class $C^1$.
Write $L_p$ for the vector of [[partial derivatives|05AXG3QRKR]] of $L$ in the variables $p$.
The *Weierstrass excess function* of $L$ is the function $E \colon [a, b] \times \R^n \times \R^n \times \R^n \to \R$ given by
$$E(x, y, p, q) = L(x, y, q) - L(x, y, p) - (q - p) \cdot L_p(x, y, p).$$

## Remarks

Example: for $L(x, y, p) = \sqrt{1 + p^2}$ with $n = 1$, the function $q \mapsto L(x, y, q)$ is convex.
A convex $C^1$ function lies above each of its tangent lines, so $E \ge 0$ everywhere.

Non-example of nonnegativity: for $L = (p^2 - 1)^2$ with $n = 1$, at $p = 0$ one gets $E(x, y, 0, q) = (q^2 - 1)^2 - 1$.
This is negative for $0 < \abs{q} < \sqrt{2}$, since $L$ is not convex in $p$.

The Weierstrass necessary condition reads as follows.
Let $y_0 \in C^1([a, b]; \R^n)$ be a strong local minimiser of $J(y) = \int_a^b L(x, y, y') \, dx$ with fixed end values, see [[weak and strong extremum|WMPCR723MJ]].
Then $E(x, y_0(x), y_0'(x), q) \ge 0$ for all $x \in [a, b]$ and all $q \in \R^n$.

For $C^2$ functions $L$, Taylor's formula gives $E(x, y, p, q) = \frac{1}{2} (q - p)^T L_{pp}(x, y, \bar p) (q - p)$ for some $\bar p$ on the segment from $p$ to $q$.
So the Weierstrass condition implies the Legendre condition $L_{pp}(x, y_0, y_0') \ge 0$.