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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 . : = \text \qquad \left \frac=p \quad;\quad \frac=q \right/math>, :\frac\cdot\frac - \left(\frac\right)^2 > 0, : is an analytic function of all its arguments and . Hilbert calls this kind of variational problem a "''regular variational problem''": property means that such kind of variational problems are minimum problems, property is the ellipticity condition on the Euler–Lagrange equations associated to the given
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
, while property is a simple regularity assumption the function . Having identified the class of problems to deal with, he then poses the following question:-"''... does every Lagrangian partial differential equation of a regular variation problem have the property of admitting analytic integrals exclusively?''" and asks further if this is the case even when the function is required to assume, as it happens for Dirichlet's problem on the potential function, boundary values which are continuous, but not analytic.


The path to the complete solution

Hilbert stated his nineteenth problem as a regularity problem for a class of elliptic partial differential equation with analytic coefficients, therefore the first efforts of the researchers who sought to solve it were directed to study the regularity of classical solutions for equations belonging to this class. For solutions Hilbert's problem was answered positively by in his thesis: he showed that solutions of nonlinear elliptic analytic equations in 2 variables are analytic. Bernstein's result was improved over the years by several authors, such as , who reduced the differentiability requirements on the solution needed to prove that it is analytic. On the other hand, direct methods in the calculus of variations showed the existence of solutions with very weak differentiability properties. For many years there was a gap between these results: the solutions that could be constructed were known to have square integrable second derivatives, which was not quite strong enough to feed into the machinery that could prove they were analytic, which needed continuity of first derivatives. This gap was filled independently by , and . They were able to show the solutions had first derivatives that were
Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
, which by previous results implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem. Subsequently, Juergen Moser gave an alternate proof of the results obtained by , and .


Counterexamples to various generalizations of the problem

The affirmative answer to Hilbert's nineteenth problem given by Ennio De Giorgi and John Forbes Nash raised the question if the same conclusion holds also for Euler-lagrange equations of more general
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
s: at the end of the 1960s, , and constructed independently several counterexamples, showing that in general there is no hope to prove such kind of regularity results without adding further hypotheses. Precisely, gave several counterexamples involving a single elliptic equation of order greater than two with analytic coefficients: for experts, the fact that such kind of equations could have nonanalytic and even nonsmooth solutions created a sensation. and gave counterexamples showing that in the case when the solution is vector-valued rather than scalar-valued, it need not be analytic: the example of De Giorgi consists of an elliptic system with bounded coefficients, while the one of Giusti and Miranda has analytic coefficients. Later on, provided other, more refined, examples for the vector valued problem.For more information about the work of Jindřich Nečas see the work of and .


De Giorgi's theorem

The key theorem proved by De Giorgi is an
a priori estimate In the theory of partial differential equations, an ''a priori'' estimate (also called an apriori estimate or ''a priori'' bound) is an estimate for the size of a solution or its derivatives of a partial differential equation. ''A priori'' is Latin ...
stating that if ''u'' is a solution of a suitable linear second order strictly elliptic PDE of the form : D_i(a^(x)\,D_ju)=0 and u has square integrable first derivatives, then u is Hölder continuous.


Application of De Giorgi's theorem to Hilbert's problem

Hilbert's problem asks whether the minimizers w of an energy functional such as :\int_UL(Dw)\,\mathrmx are analytic. Here w is a function on some compact set U of R''n'', Dw is its
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
vector, and L is the Lagrangian, a function of the derivatives of w that satisfies certain growth, smoothness, and convexity conditions. The smoothness of w can be shown using De Giorgi's theorem as follows. The Euler–Lagrange equation for this variational problem is the non-linear equation : \sum\limits_^n(L_(Dw))_ = 0 and differentiating this with respect to x_k gives : \sum\limits_^n(L_(Dw)w_)_ = 0 This means that u=w_ satisfies the linear equation : D_i(a^(x)D_ju)=0 with :a^ = L_(Dw) so by De Giorgi's result the solution ''w'' has Hölder continuous first derivatives, provided the matrix L_ is bounded. When this is not the case, a further step is needed: one must prove that the solution w is Lipschitz continuous, i.e. the gradient Dw is an L^\infty function. Once ''w'' is known to have Hölder continuous (''n''+1)st derivatives for some ''n'' ≥ 1, then the coefficients ''a''''ij'' have Hölder continuous ''n''th derivatives, so a theorem of Schauder implies that the (''n''+2)nd derivatives are also Hölder continuous, so repeating this infinitely often shows that the solution ''w'' is smooth.


Nash's theorem

Nash gave a continuity estimate for solutions of the parabolic equation : D_i(a^(x)D_ju)=D_t(u) where ''u'' is a bounded function of ''x''1,...,''x''''n'', ''t'' defined for ''t'' ≥ 0. From his estimate Nash was able to deduce a continuity estimate for solutions of the elliptic equation : D_i(a^(x)D_ju)=0 by considering the special case when ''u'' does not depend on ''t''.


Notes


References

*. *. Reprinted in . *. "''On the analyticity of extremals of multiple integrals''" (English translation of the title) is a short research announcement disclosing the results detailed later in . While, according to the Complete list of De Giorgi's scientific publication (De Giorgi 2006, p. 6), an English translation should be included in , it is unfortunately missing. *. Translated in English as "''On the differentiability and the analyticity of extremals of regular multiple integrals''" in . *. Translated in English as "''An example of discontinuous extremals for a variational problem of elliptic type''" in . *. *. *. *, translated in English as . *. *. * *.
– Reprinted as .
– Translated to English by
Mary Frances Winston Newson Mary Frances Winston Newson (August 7, 1869 December 5, 1959) was an American mathematician. She became the first female American to receive a PhD in mathematics from a European university, namely the University of Göttingen in Germany.Grinstein ...
as .
– Reprinted as .
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as .
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as . *. *.
– Translated in English as . *. *. *. *. *. *. {{Hilbert's problems #19 Partial differential equations Calculus of variations