Calculus of variations and optimal control; optimization Second variation and sufficient conditions Theorem

Legendre–Hadamard condition

Builds on Integral functional with a Lagrangian density Weak and strong extremum Second variation

Statement

Let $\Omega \subseteq \R^n$ be a bounded open set and let $m \ge 1$ . Let $L \colon \overline{\Omega} \times \R^m \times \R^{m \times n} \to \R$ , written $L(x, u, \xi)$ with $\xi = (\xi^i_\alpha)$ , be of class $C^2$ . Let $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ be the integral functional on $C^1(\overline{\Omega}; \R^m)$ . Let $u_0 \in C^1(\overline{\Omega}; \R^m)$ be a weak local minimiser of $J$ among functions with the same boundary values. Then for every $x \in \Omega$ , every $a \in \R^m$ and every $b \in \R^n$ , $\sum_{i, j = 1}^{m} \sum_{\alpha, \beta = 1}^{n} \frac{\partial^2 L}{\partial \xi^i_\alpha \, \partial \xi^j_\beta}\big(x, u_0(x), \nabla u_0(x)\big) \, a_i a_j b_\alpha b_\beta \ge 0.$

Remarks

The second variation at $u_0$ is nonnegative for all test functions $\varphi \in C^1_c(\Omega; \R^m)$ . One tests it with $\varphi(x) = \varepsilon \, \eta(x) \, a \sin(b \cdot x / \varepsilon)$ for a fixed smooth cut-off function $\eta$ supported near $x_0$ . As $\varepsilon \to 0$ , only the term quadratic in $\nabla \varphi$ survives, and its limit is a positive multiple of the stated quadratic form at $x_0$ weighted by $\eta^2$ .

The condition asks for positivity only on rank-one matrices $a \otimes b = (a_i b_\alpha)$ , not on all of $\R^{m \times n}$ . For $m = 1$ or $n = 1$ every matrix is of the form $a \otimes b$ . Then the condition says that the Hessian of $L$ in $\xi$ is positive semidefinite, as in the one-dimensional Legendre condition.

Example: for $m = n = 2$ and $L(\xi) = \det \xi$ , the quadratic form is $2 \det(a \otimes b) = 0$ , so the condition holds although the Hessian of $\det$ is indefinite. For a $C^2$ integrand, the condition at every $\xi$ is equivalent to rank-one convexity of $\xi \mapsto L(x, u, \xi)$ .

Going deeper

For a student

The Legendre–Hadamard condition is the necessary condition from the second variation for vector-valued problems in several variables. When it holds with a strict inequality and a uniform constant, the Euler–Lagrange system is called strongly elliptic, which is the basic hypothesis of linear elliptic theory for systems. It is the infinitesimal form of rank-one convexity, which in turn is implied by Morrey's quasiconvexity.

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Integral functional with a Lagrangian density concept
Second variation concept
Weak and strong extremum concept
Legendre–Hadamard condition statement

Tag	`JMVHZ6HDNZ`
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About	Second variation
Uses	Integral functional with a Lagrangian density, Weak and strong extremum, Second variation