Hilbert's nineteenth problem

{{quote

|text= Eine der begrifflich merkwürdigsten Thatsachen in den Elementen der Theorie der analytischen Funktionen erblicke ich darin, daß es Partielle Differentialgleichungen giebt, deren Integrale sämtlich notwendig analytische Funktionen der unabhängigen Variabeln sind, die also, kurz gesagt, nur analytischer Lösungen fähig sind.English translation by Mary Frances Winston Newson:-"One of the most remarkable facts in the elements of the theory of analytic functions appears to me to be this: that there exist partial differential equations whose integrals are all of necessity analytic functions of the independent variables, that is, in short, equations susceptible of none but analytic solutions".

|sign= David Hilbert

|source= {{harv|Hilbert|1900|p=288}}.

}}

David Hilbert presented what is now called his nineteenth problem in his speech at the second International Congress of Mathematicians.For a detailed historical analysis, see the relevant entry "Hilbert's problems". In {{harv|Hilbert|1900|p=288}} 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 analytic functions as solutions, listing Laplace's equation, Liouville's equation,Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature {{math|K}} as equal to {{math|-1/2}}: compare the relevant entry with {{harv|Hilbert|1900|p=288}}. the minimal surface equation and a class of linear partial differential equations studied by Émile Picard as examples.Unlike Liouville's work, Picard's work is explicitly cited by {{harvtxt|Hilbert|1900|loc=p. 288 and footnote 1 in the same page}}. He then notes that most partial differential equations sharing this property are Euler–Lagrange equations of a well defined kind of variational problem, satisfying the following three properties:See {{harv|Hilbert|1900|p=288}}.

:{{EquationRef|1|(1){{spaces|5}}}}$\{\backslash iint\; F(p,q,z;x,y)\; dx\; dy\}\; =\; \backslash text\{Minimum\}\; \backslash qquad$

\left[ \frac{\partial z}{\partial x}=p \quad;\quad \frac{\partial z}{\partial y}=q \right],

:{{EquationRef|2|(2){{spaces|5}}}}$\backslash frac\{\backslash partial^2\; F\}\{\backslash partial^2\; p\}\backslash cdot\backslash frac\{\backslash partial^2\; F\}\{\backslash partial^2\; q\}\; -\; \backslash left(\backslash frac\{\backslash partial^2\; F\}\{\{\backslash partial\; p\}\{\backslash partial\; q\}\}\backslash right)^2\; >\; 0$,

:{{EquationRef|3|(3){{spaces|5}}}} {{math|F}} is an analytic function of all its arguments {{math|p, q, z, x}} and {{math|y}}.

Hilbert calls this a "regular variational problem".In his exact words: "Reguläres Variationsproblem". Hilbert's definition of a regular variational problem is stronger than the one currently used, for example, in {{harv|Gilbarg|Trudinger|2001|p=289}}. Property {{EquationNote|(1)}} means that these are minimum problems. Property {{EquationNote|(2)}} is the ellipticity condition on the Euler–Lagrange equations associated to the given functional, while property {{EquationNote|(3)}} is a simple regularity assumption about the function {{math|F}}.Since Hilbert considers all derivatives in the "classical", i.e. not in the weak but in the strong, sense, even before the statement of its analyticity in {{EquationNote|(3)}}, the function {{math|F}} is assumed to be at least {{math|{{SubSup|C||2}}}}, as the use of the Hessian determinant in {{EquationNote|(2)}} implies. Having identified the class of problems considered, he poses the following question: "... does every Lagrangian partial differential equation of a regular variation problem have the property of admitting analytic integrals exclusively?"English translation by Mary Frances Winston Newson: #{{harvid, p. 288) precise words are:-"... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt" (Italics emphasis by Hilbert himself). He asks further if this is the case even when the function is required to assume boundary values that are continuous, but not analytic, as happens for Dirichlet's problem for the potential function .

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 researchers who sought to solve it were aimed at studying the regularity of classical solutions for equations belonging to this class. For {{math|Continuously differentiable function}} solutions, Hilbert's problem was answered positively by {{harvs|txt|first=Sergei|last=Bernstein|authorlink=Sergei Natanovich Bernstein|year=1904}} in his thesis. He showed that {{math|{{SubSup|C||3}}}} solutions of nonlinear elliptic analytic equations in two variables are analytic. Bernstein's result was improved over the years by several authors, such as {{harvtxt|Petrowsky|1939}}, 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, but this 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 {{harvs|txt|author-link=Ennio De Giorgi|first=Ennio |last= De Giorgi|year1=1956|year2= 1957}}, and {{harvs|txt|first=John Forbes |last=Nash|author-link=John Forbes Nash|year1=1957|year2=1958}}, who were able to show the solutions had first derivatives that were Hölder continuous. By previous results this implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem. Subsequently, Jürgen Moser gave an alternate proof of the results obtained by {{harvs|txt|author-link=Ennio De Giorgi|first=Ennio |last= De Giorgi|year1=1956|year2= 1957}}, and {{harvs|txt|first=John Forbes |last=Nash|author-link=John Forbes Nash|year1=1957|year2=1958}}.

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 functionals. At the end of the 1960s, {{harvtxt|Maz'ya|1968}},See {{harv|Giaquinta|1983|p=59}}, {{harv|Giusti|1994|loc=p. 7 footnote 7 and p. 353}}, {{harv|Gohberg|1999|p=1}}, {{harv|Hedberg|1999|pp=10–11}}, {{harv|Kristensen|Mingione|2011|loc=p. 5 and p. 8}}, and {{harv|Mingione|2006|p=368}}. {{harvtxt|De Giorgi|1968}} and {{harvtxt|Giusti|Miranda|1968}} independently constructed several counterexamples,See {{harv|Giaquinta|1983|pp=54–59}}, {{harv|Giusti|1994|loc=p. 7 and pp. 353}}. showing that in general there is no hope of proving such regularity results without adding further hypotheses.

Precisely, {{harvtxt|Maz'ya|1968}} gave several counterexamples involving a single elliptic equation of order greater than two with analytic coefficients.See {{harv|Hedberg|1999|pp=10–11}}, {{harv|Kristensen|Mingione|2011|loc=p. 5 and p. 8}} and {{harv|Mingione|2006|p=368}}. For experts, the fact that such equations could have nonanalytic and even nonsmooth solutions created a sensation.According to {{harv|Gohberg|1999|p=1}}.

{{harvtxt|De Giorgi|1968}} and {{harvtxt|Giusti|Miranda|1968}} 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.See {{harv|Giaquinta|1983|pp=54–59}} and {{harv|Giusti|1994|loc=p. 7, pp. 202–203 and pp. 317–318}}. Later, {{harvtxt|Nečas|1977}} provided other, more refined, examples for the vector valued problem.For more information about the work of Jindřich Nečas see the work of {{harvtxt|Kristensen|Mingione|2011|loc=§3.3, pp. 9–12}} and {{harv|Mingione|2006|loc=§3.3, pp. 369–370}}.

The key theorem proved by De Giorgi is an a priori estimate stating that if u is a solution of a suitable linear second order strictly elliptic PDE of the form

:$D\_i(a^\{ij\}(x)\backslash ,D\_ju)=0$

and $u$ has square integrable first derivatives, then $u$ is Hölder continuous.

Hilbert's problem asks whether the minimizers $w$ of an energy functional such as

:$\backslash int\_UL(Dw)\backslash ,\backslash mathrm\{d\}x$

are analytic. Here $w$ is a function on some compact set $U$ of Rⁿ, $Dw$ is its gradient 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

:$\backslash sum\backslash limits\_\{i=1\}^n(L\_\{p\_i\}(Dw))\_\{x\_i\}\; =\; 0$

and differentiating this with respect to $x\_k$ gives

:$\backslash sum\backslash limits\_\{i=1\}^n(L\_\{p\_ip\_j\}(Dw)w\_\{x\_jx\_k\})\_\{x\_i\}\; =\; 0$

This means that $u=w\_\{x\_k\}$ satisfies the linear equation

:$D\_i(a^\{ij\}(x)D\_ju)=0$

with

:$a^\{ij\}\; =\; L\_\{p\_ip\_j\}(Dw)$

so by De Giorgi's result the solution w has Hölder continuous first derivatives, provided the matrix $L\_\{p\_ip\_j\}$ 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 $L^\backslash infty$ 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 nth 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.

John Nash gave a continuity estimate for solutions of the parabolic equation

:$D\_i(a^\{ij\}(x)D\_ju)=D\_t(u)$

where u is a bounded function of x₁,...,x_n, t defined for t ≥ 0. From his estimate Nash was able to deduce a continuity estimate for solutions of the elliptic equation

:$D\_i(a^\{ij\}(x)D\_ju)=0$ by considering the special case when u does not depend on t.

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| first = Lars Inge

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| contribution =On Maz'ya's work in potential theory and the theory of function spaces

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| year =1999

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| year = 1900

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| url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1900&DMDID=DMDLOG_0037

| jfm =31.0068.03

}}.
– Reprinted as {{Citation|ref=none

| title = Mathematische Probleme

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| year = 1900

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| jfm = 32.0084.05

}}.
– Translated to English by Mary Frances Winston Newson as {{Citation|ref=none

| last = Hilbert

| first = David

| author-link = David Hilbert

| title = Mathematical Problems

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| year = 1902

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| jfm = 33.0976.07

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}}.
– Reprinted as {{Citation|ref=none

| last = Hilbert

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| author-link = David Hilbert

| title = Mathematical Problems

| journal = Bulletin of the American Mathematical Society

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| volume = 37

| issue = 4

| pages = 407–436

| year = 2000

| doi = 10.1090/S0273-0979-00-00881-8

| mr = 1779412

| zbl = 0979.01028

| s2cid = 12695502

| doi-access = free

}}.
– Translated to French by M. L. Laugel (with additions of Hilbert himself) as {{Citation|ref=none

| last = Hilbert

| first = David

| author-link = David Hilbert

| editor-last = Duporcq

| editor-first = E.

| contribution = Sur les problèmes futurs des Mathématiques

| contribution-url = http://www.mathunion.org/ICM/ICM1900/Main/icm1900.0058.0114.ocr.pdf

| title = Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications

| series = ICM Proceedings

| year = 1902

| pages = 58–114

| place = Paris

| publisher = Gauthier-Villars

| url = http://www.mathunion.org/ICM/ICM1900/

| jfm = 32.0084.06

| access-date = 2013-12-28

| archive-url = https://web.archive.org/web/20131231002929/http://www.mathunion.org/ICM/ICM1900/

| archive-date = 2013-12-31

| url-status = dead

}}.
– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as {{Citation|ref=none

| last = Hilbert

| first = D.

| author-link = David Hilbert

| title =Problèmes mathématiques

| journal =L'Enseignement Mathématique

| volume =2

| pages =349–355

| year =1900

| language =fr

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|id = OxPDE-11/17

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|date = October 2011

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| year =1968

| language = ru

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– Translated in English as {{Citation|ref=none

| last = Maz'ya

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| author-link = Vladimir Gilelevich Maz'ya

| title = Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients

| journal = Functional Analysis and Its Applications

| volume = 2

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| pages = 230–234

| year =1968

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| mr =2291779

| zbl =1164.49324

| doi=10.1007/s10778-006-0110-3

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| place = Berlin–Heidelberg–New York

| publisher = Springer-Verlag

| series = Die Grundlehren der mathematischen Wissenschaften

| volume=130

| year=1966

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}}.

• {{Citation

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| author-link=John Forbes Nash

| title=Parabolic equations

| journal=Proceedings of the National Academy of Sciences of the United States of America

| year=1957

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| issue=8

| pages=754–758

| issn=0027-8424

| jstor=89599

| mr=0089986

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| doi=10.1073/pnas.43.8.754

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}}.

• {{Citation

| last1=Nash

| first1=John

| author1-link=John Forbes Nash

| title=Continuity of solutions of parabolic and elliptic equations

| year=1958

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| volume=80

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| pages=931–954

| issn=0002-9327

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| url=http://dml.cz/bitstream/handle/10338.dmlcz/101876/CzechMathJ_33-1983-2_7.pdf

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| first =Jindřich

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| editor2-first =Wolfdietrich

| contribution = Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity

| title = Theory of nonlinear operators: constructive aspects. Proceedings of the fourth international summer school, held at Berlin, GDR, from September 22 to 26, 1975

| series = Abhandlungen der Akademie der Wissenschaften der DDR

| volume = 1

| year = 1977

| pages = 197–206

| place = Berlin

| publisher = Akademie-Verlag

| mr =0509483

| zbl=0372.35031

}}.

• {{Citation

| last1=Petrowsky

| first1=I. G.

| author-link=Ivan Georgievich Petrovsky

| title= Sur l'analyticité des solutions des systèmes d'équations différentielles

| url= http://mi.mathnet.ru/eng/msb5769

| year=1939

| language = fr

| journal = Recueil Mathématique (Matematicheskii Sbornik)

| volume = 5

| issue = 47

| pages = 3–70

| jfm = 65.0405.02

| mr = 0001425

| zbl = 0022.22601

}}.

{{Hilbert's problems}}

#19

Category:Partial differential equations

Category:Calculus of variations