Hilbert's nineteenth problem is one of the 23
Hilbert problems, set out in a list compiled in 1900 by
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
. It asks whether the solutions of regular problems in the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
are always
analytic
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
. Informally, and perhaps less directly, since Hilbert's concept of a "''regular variational problem''" identifies precisely a
variational problem
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
whose
Euler–Lagrange equation is an
elliptic partial differential equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
whe ...
with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s, any solution function inherits the relatively simple and well understood structure from the solved equation. Hilbert's nineteenth problem was solved independently in the late 1950s by
Ennio De Giorgi and
John Forbes Nash, Jr.
History
The origins of the problem
David Hilbert presented the now called Hilbert's nineteenth problem in his speech at the second
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rena ...
. In he states that, in his opinion, one of the most remarkable facts of the theory of analytic functions is that there exist classes of partial differential equations which admit only such kind of functions as solutions, adducing
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \na ...
,
Liouville's equation
:''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).''
: ''For Liouville's equation in quantum mechanics, see Von Neumann equation.''
: ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...
, the
minimal surface equation
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
and a class of linear partial differential equations studied by
Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at ...
as examples. He then notes the fact that most of the partial differential equations sharing this property are the Euler–Lagrange equation of a well defined kind of variational problem, featuring the following three properties:
[See .]
: