Calculus of variations and optimal control; optimization Direct methods and existence of minimizers Theorem

Tonelli's existence theorem

Builds on Integral functional with a Lagrangian density Sobolev space Convex integrand Coercive functional Weak lower semicontinuity Direct method in the calculus of variations

Statement

Let $\Omega \subseteq \R^n$ be a bounded open set with Lipschitz boundary, and let $1 < p < \infty$ . Let $L \colon \overline{\Omega} \times \R \times \R^n \to \R$ , written $L(x, u, \xi)$ , be continuous. Assume:

$L$ is a convex integrand: $\xi \mapsto L(x, u, \xi)$ is convex for all $(x, u)$ ;
there are constants $a > 0$ and $b \in \R$ with $L(x, u, \xi) \ge a \abs{\xi}^p + b$ for all $(x, u, \xi)$ .

Let $u_0$ be in the Sobolev space $W^{1,p}(\Omega)$ and let $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ , an integral functional. If $J(u_0) < +\infty$ , then $J$ attains its minimum on $u_0 + W^{1,p}_0(\Omega)$ .

Remarks

The proof is the direct method. The growth bound and the Poincaré inequality make $J$ coercive on $u_0 + W^{1,p}_0(\Omega)$ , so a minimising sequence is bounded in $W^{1,p}(\Omega)$ . Since $1 < p < \infty$ , a subsequence converges weakly in $W^{1,p}(\Omega)$ , and strongly in $L^p(\Omega)$ by the Rellich–Kondrachov theorem. Convexity in $\xi$ makes $J$ weakly lower semicontinuous, so the weak limit is a minimiser.

Example: for $L = \frac{1}{2} \abs{\xi}^2 + \frac{1}{4} u^4$ and $p = 2$ , the theorem gives a minimiser with boundary values $u_0$ whenever $J(u_0) < +\infty$ ; it is a weak solution of $-\Delta u + u^3 = 0$ .

Each hypothesis matters. For $L = (\xi^2 - 1)^2 + u^2$ with $n = 1$ and $u_0 = 0$ , convexity fails and there is no minimiser. For $L = \sqrt{1 + \abs{\xi}^2}$ , which has growth $p = 1$ , minimisers of the area of a graph need not exist in $W^{1,1}$ for some boundary data.

Going deeper

For a student

This is the model existence theorem of the calculus of variations, and it is the template for the direct method. The minimiser is a priori only a Sobolev function; whether it is smooth is the separate question of regularity, raised in Hilbert's nineteenth problem. For vector-valued $u$ , convexity can be weakened to Morrey quasiconvexity, and in nonlinear elasticity to polyconvexity, as in Ball's existence theorem. Without convexity one replaces the integrand by its convex envelope; this is relaxation, and it computes the limits of minimising sequences.

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Coercive functional concept
Convex integrand concept
Direct method in the calculus of variations concept
Integral functional with a Lagrangian density concept
Sobolev space concept
Weak lower semicontinuity concept
Tonelli's existence theorem statement

Tag	`NJQRG3DVPP`
URI	`m2://NJQRG3DVPP`
Kind	statement, theorem
Writing	W0 stub
MSC	49J10, 49J45
Status	This copy of the site has no build data. Run `m2 status NJQRG3DVPP`.
About	Direct method in the calculus of variations
Uses	Integral functional with a Lagrangian density, Sobolev space, Convex integrand, Coercive functional, Weak lower semicontinuity, Direct method in the calculus of variations

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