---
tag: 187NP5Z002
kind: problem
name: Hilbert's twentieth problem
msc: [49J10, 35A15, 35J20]
uses: [PQ25NHV0KS, TE379HBTY0, TM8VQJWGHZ]
about: [PQ25NHV0KS]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_twentieth_problem
same_as:
  wikidata: Q5761174
  wikipedia: Hilbert's twentieth problem
history:
  posed_by: "David Hilbert"
  first_stated: "1900"
---

## Problem

Let $\Omega \subseteq \R^n$ be a bounded open set.
Consider a regular problem of the [[calculus of variations|PQ25NHV0KS]]: 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|TM8VQJWGHZ]] 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|TE379HBTY0]] 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|QPMM2ZPWBW]], 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.

## Explanations

### L1 human

Hilbert's twentieth problem asks whether every reasonable minimisation problem with boundary values has a solution, if one is willing to widen what counts as a solution.
The twentieth century answered this mostly in the affirmative, by working with weak solutions in Sobolev spaces and then proving that they are in fact smooth.