Calculus of variations and optimal control; optimization Problem

Hilbert's twentieth problem

Builds on Calculus of variations Direct method in the calculus of variations Boundary value problem

Problem

Let $\Omega \subseteq \R^n$ be a bounded open set. Consider a regular problem of the calculus of variations: minimise $J(u) = \int_\Omega L(x, u, \nabla u) \, dx$ with given values of $u$ on $\partial\Omega$ . Here the Lagrangian $L$ is smooth and strictly convex in $\nabla u$ . Hilbert asked whether every such problem has a solution. He allowed suitable assumptions on the boundary conditions and, if needed, a suitable extension of the notion of solution.

Remarks

The problem is a programme rather than a single statement, and its answer depends on how "suitable" is made precise. The modern answer is largely affirmative. The direct method gives minimisers in Sobolev spaces for Lagrangians that are convex in $\nabla u$ and satisfy growth conditions, which is the "suitably extended" notion of solution. Regularity theory, which grew from Hilbert's nineteenth problem, then shows that these generalised solutions are classical in many cases. The same circle of ideas gives weak solutions of boundary value problems for linear and nonlinear elliptic equations.

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Boundary value problem concept
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Hilbert's twentieth problem problem

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Hilbert's twenty-third problem

Tag	`187NP5Z002`
URI	`m2://187NP5Z002`
Kind	problem
Writing	W0 stub
MSC	49J10, 35A15, 35J20
Status	This copy of the site has no build data. Run `m2 status 187NP5Z002`.
About	Calculus of variations
Uses	Calculus of variations, Direct method in the calculus of variations, Boundary value problem
Same as	Wikidata Q5761174 Wikipedia Hilbert's twentieth problem

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Hilbert's twenty-third problem