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Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a celebrated list compiled in 1900 by David Hilbert.[1] It asks whether the solutions of regular problems in the calculus of variations are always analytic.[2] Informally, and perhaps less directly, since Hilbert's concept of a "regular variational problem" identifies precisely a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients.[3] Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of partial differential equations, any solution function inherits the relatively simple and well understood structure from the solved equation.

History
The origins of the problem

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.[4]
—David Hilbert, (Hilbert 1900, p. 288).

David Hilbert presented the now called Hilbert's nineteen problem in his speech at the second International Congress of Mathematicians.[5] In (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 such kind of functions as solutions, adducing Laplace's equation, Liouville's equation,[6] the minimal surface equation and a class of linear partial differential equations studied by Émile Picard as examples.[7] 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:[8]

(1) $${\iint F(p,q,z;x,y) dx dy} = \text{Minimum} \qquad \left[ \frac{\partial z}{\partial x}=p \quad;\quad \frac{\partial z}{\partial y}=q \right],$$
(2) $$\frac{\partial^2 F}{\partial^2 p}\cdot\frac{\partial^2 F}{\partial^2 q} - \left(\frac{\partial^2 F}{{\partial p}{\partial q}}\right)^2 > 0,$$
(3) F is an analytic function of all its arguments p, q, z, x and y.

Hilbert calls this kind of variational problem a "regular variational problem":[9] property (1) means that such kind of variational problems are minimum problems, property (2) is the ellipticity condition on the Euler–Lagrange equations associated to the given functional, while property (3) is a simple regularity assumption the function F.[10] 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?"[11] 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.[8]
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,[8] 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 C 3 solutions Hilbert's problem was answered positively by Sergei Bernstein (1904) in his thesis: he showed that C 3 solutions of nonlinear elliptic analytic equations in 2 variables are analytic. Bernstein's result was improved over the years by several authors, such as 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, 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 Ennio De Giorgi (1956, 1957), and John Forbes Nash (1957, 1958). They were able to show the solutions had first derivatives that were Hölder continuous, 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.
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 functionals: at the end of the sixties, Maz'ya (1968),[12] De Giorgi (1968) and Giusti & Miranda (1968) constructed independently several counterexamples,[13] showing that in general there is no hope to prove such kind of regularity results without adding further hypotheses.

Precisely, Maz'ya (1968) gave several counterexamples involving a single elliptic equation of order greater than two with analytic coefficients:[14] for experts, the fact that such kind of equations could have nonanalytic and even nonsmooth solutions created a sensation.[15]

De Giorgi (1968) and Giusti & Miranda (1968) gave counterexamples showing that in the case when the solution is vector-valued rather than scalar-valued, it need not to 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.[16] Later on, Nečas (1977) provided other, more refined, examples for the vector valued problem.[17]
De Giorgi's theorem

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

$$D_i(a^{ij}(x)D_ju)=0$$

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

$$\int_UL(Dw)\mathrm{d}x$$

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

$$\Sigma_i(L_{p_i}(Dw))_{x_i} = 0$$

and differentiating this with respect to xk gives

$$\Sigma_i(L_{p_ip_j}(Dw)w_{x_jx_k})_{x_i} = 0$$

This means that u=wxk 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.

Once w is known to have Hölder continuous (n+1)st derivatives for some n ≥ 0, then the coefficients aij 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.
Nash's theorem

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,...,xn, 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.

Notes

See (Hilbert 1900) or, equivalently, one of its translations.

"Sind die Lösungen regulärer Variationsprobleme stets notwending analytisch?" (English translation by Mary Frances Winston Newson:-"Are the solutions of regular problems in the calculus of variations always necessarily analytic?"), formulating the problem with the same words of Hilbert (1900, p. 288).
See (Hilbert 1900, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translation or reprint, or the subsection "The origins of the problem" in the historical section of this entry.
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".
For a detailed historical analysis, see the relevant entry "Hilbert's problems".
Hilbert does not cite explicitly Joseph Liouville and considers the constant Gaussian curvature K as equal to -1/2: compare the relevant entry with (Hilbert 1900, p. 288).
Contrary to Liouville's work, Picard's work is explicitly cited by Hilbert (1900, p. 288 and footnote 1 in the same page).
See (Hilbert 1900, p. 288).
"Reguläres Variationsproblem", in his exact words. Hilbert's definition of a regular variational problem is stronger than the currently used one, found, for example, in (Gilbarg & Trudinger 2001, p. 289).
Since Hilbert considers all derivatives in the "classical", i.e. not in the weak nor in the strong, sense, even before the statement of its analyticity in (3), the function F is assumed to be at least C 2 , as the use of the Hessian determinant in (2) implies.
English translation by Mary Frances Winston Newson: Hilbert's (1900, 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).
See (Giaquinta 1983, p. 59), (Giusti 1994, p. 7 footnote 7 and p. 353), (Gohberg 1999, p. 1), (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8), and (Mingione 2006, p. 368).
See (Giaquinta 1983, pp. 54–59), (Giusti 1994, p. 7 and pp. 353).
See (Hedberg 1999, pp. 10–11), (Kristensen & Mingione 2011, p. 5 and p. 8) and (Mingione 2006, p. 368).
According to (Gohberg 1999, p. 1).
See (Giaquinta 1983, pp. 54–59) and (Giusti 1994, p. 7, pp. 202–203 and pp. 317–318).

For more information about the work of Jindřich Nečas see the work of Kristensen & Mingione (2011, §3.3, pp. 9–12) and (Mingione 2006, §3.3, pp. 369–370).

Hilbert's problems : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

References

Bernstein, S. (1904), "Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre", Mathematische Annalen (in French) 59: 20–76, doi:10.1007/BF01444746, ISSN 0025-5831, JFM 35.0354.01.
Bombieri, Enrico (1975), "Variational problems and elliptic equations" (PDF), Proceedings of the International Congress of Mathematicians, Vancouver, B.C., 1974, Vol. 1, ICM Proceedings, Montreal: Canadian Mathematical Congress, pp. 53–63, MR 0509259, Zbl 0344.49002. Reprinted in Bombieri, Enrico (1976), "Variational problems and elliptic equations", in Browder, Felix E., Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics, XXVIII, Providence, Rhode Island: American Mathematical Society, pp. 525–535, ISBN 978-0-8218-1428-4, MR 0425740, Zbl 0347.35032.
De Giorgi, Ennio (1956), "Sull'analiticità delle estremali degli integrali multipli", Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII, (in Italian) 20: 438–441, MR 0082045, Zbl 0074.31503. "On the analyticity of extremals of multiple integrals" (English translation of the title) is a short research announcement disclosing the results detailed later in (De Giorgi 1957). While, according to the Complete list of De Giorgi's scientific publication (De Giorgi 2006, p. 6), an English translation should be included in (De Giorgi 2006), it is unfortunately missing.
De Giorgi, Ennio (1957), "Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari", Memorie della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematicahe e Naturali., Serie III, (in Italian) 3: 25–43, MR 0093649, Zbl 0084.31901. Translated in English as "On the differentiability and the analyticity of extremals of regular multiple integrals" in (De Giorgi 2006, pp. 149–166).
De Giorgi, Ennio (1968), "Un esempio di estremali discontinue per un problema variazionale di tipo ellittico", Bollettino dell'Unione Matematica Italiana (4), Serie IV, (in Italian) 1: 135–137, MR 0227827, Zbl 0084.31901. Translated in English as "An example of discontinuous extremals for a variational problem of elliptic type" in (De Giorgi 2006, pp. 285–287).
De Giorgi, Ennio (2006), Ambrosio, Luigi; Dal Maso, Gianni; Forti, Marco; Miranda, Mario; Spagnolo, Sergio, eds., Selected papers, Berlin–New York: Springer-Verlag, pp. x+889, ISBN 978-3-540-26169-8, MR 2229237, Zbl 1096.01015.
Giaquinta, Mariano (1983), Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105, Princeton, New Jersey: Princeton University Press, pp. vii+297, ISBN 0-691-08330-4, MR 0717034, Zbl 0516.49003.
Gilbarg, David; Trudinger, Neil S. (2001) [1998], Elliptic partial differential equations of second order, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517, ISBN 3-540-41160-7, MR 1814364, Zbl 1042.35002.
Giusti, Enrico (1994), Metodi diretti nel calcolo delle variazioni, Monografie Matematiche (in Italian), Bologna: Unione Matematica Italiana, pp. VI+422, MR 1707291, Zbl 0942.49002, translated in English as Direct Methods in the Calculus of Variations, River Edge, New Jersey – London – Singapore: World Scientific Publishing, 2003, pp. viii+403, ISBN 981-238-043-4, MR 1962933, Zbl 1028.49001.
Giusti, Enrico; Miranda, Mario (1968), "Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni", Bollettino dell'Unione Matematica Italiana, Serie IV, (in Italian) 2: 1–8, MR 0232265, Zbl 0155.44501.
Gohberg, Israel (1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther, The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998, Operator Theory. Advances and Applications 109, Basel: Birkhäuser Verlag, pp. 1–5, ISBN 978-3-7643-6201-0, MR 1747861, Zbl 0939.01018.
Hedberg, Lars Inge (1999), "On Maz'ya's work in potential theory and the theory of function spaces", in Rossmann, Jürgen; Takáč, Peter; Wildenhain, Günther, The Maz'ya Anniversary Collection. Volume 1: On Maz'ya's work in functional analysis, partial differential equations and applications, 109, Operator Theory: Advances and Applicationsǘ, Basel: Birkhäuser Verlag, pp. 7–16, doi:10.1007/978-3-0348-8675-8_2, MR 1747862, Zbl 0939.31001
Hilbert, David (1900), "Mathematische Probleme", Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German) (3): 253–297, JFM 31.0068.03 (reprinted as "Mathematische Probleme", Archiv der Mathematik und Physik, dritte reihe (in German) 1, 1900: 44–63 and 253–297, JFM 32.0084.05), translated in English by Mary Frances Winston Newson as Hilbert, David (1902), "Mathematical Problems", Bulletin of the American Mathematical Society 8 (10): 437–479, doi:10.1090/S0002-9904-1902-00923-3, JFM 33.0976.07, MR 1557926 (reprinted as Hilbert, David (2000), "Mathematical Problems", Bulletin of the American Mathematical Society, New Series 37 (4): 407–436, doi:10.1090/S0273-0979-00-00881-8, MR 1779412, Zbl 0979.01028), and in French (with additions of Hilbert himself) by M. L. Laugel as Hilbert, David (1902), "Sur les problèmes futurs des Mathématiques" (PDF), in Duporcq, E., Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114, JFM 32.0084.06. There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as Hilbert, D. (1900), "Problèmes mathématiques", L'Enseignement Mathématique (in French) 2: 349–355, doi:10.5169/seals-3575, JFM 31.0905.03.
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