“Hilbert's nineteenth problem”的意思、由来-开放百科全书

Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a 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

David Hilbert presented the now called Hilbert's nineteenth problem in his speech at the second International Congress of Mathematicians.^[5] 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 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]

Hilbert calls this kind of variational problem a "regular variational problem":^[9] property {{EquationNote|(1)}} means that such kind of variational problems 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 the function {{math|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 {{math|{{SubSup|C||3}}}} 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 2 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, 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 {{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}}. 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, {{harvtxt|Maz'ya|1968}},^[12] {{harvtxt|De Giorgi|1968}} and {{harvtxt|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, {{harvtxt|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'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

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

are analytic. Here

is a function on some compact set

of Rⁿ,

is its gradient vector, and

is the Lagrangian, a function of the derivatives of

that satisfies certain growth, smoothness, and convexity conditions. The smoothness of

can be shown using De Giorgi's theorem

as follows. The Euler–Lagrange equation for this variational problem is the non-linear equation

so by De Giorgi's result the solution w has Hölder continuous first derivatives, provided the matrix

is bounded. When this is not the case, a further step is needed: one must prove that the solution

is Lipschitz continuous, i.e. the gradient

is an

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.

Nash's theorem

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

Notes

1. ^See {{harv|Hilbert|1900}} or, equivalently, one of its translations.
2. ^"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 {{harvtxt|Hilbert|1900|p=288}}.
3. ^See {{harv|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.
4. ^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".
5. ^For a detailed historical analysis, see the relevant entry "Hilbert's problems".
6. ^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}}.
7. ^Contrary to Liouville's work, Picard's work is explicitly cited by {{harvtxt|Hilbert|1900|loc=p. 288 and footnote 1 in the same page}}.
8. ^¹²See {{harv|Hilbert|1900|p=288}}.
9. ^"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 {{harv|Gilbarg|Trudinger|2001|p=289}}.
10. ^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.
11. ^English translation by Mary Frances Winston Newson: Hilbert|1900}}|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).
12. ^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}}.
13. ^See {{harv|Giaquinta|1983|pp=54–59}}, {{harv|Giusti|1994|loc=p. 7 and pp. 353}}.
14. ^See {{harv|Hedberg|1999|pp=10–11}}, {{harv|Kristensen|Mingione|2011|loc=p. 5 and p. 8}} and {{harv|Mingione|2006|p=368}}.
15. ^According to {{harv|Gohberg|1999|p=1}}.
16. ^See {{harv|Giaquinta|1983|pp=54–59}} and {{harv|Giusti|1994|loc=p. 7, pp. 202–203 and pp. 317–318}}.
17. ^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}}.

References

}}. "On the analyticity of extremals of multiple integrals" (English translation of the title) is a short research announcement disclosing the results detailed later in {{harv|De Giorgi|1957}}. While, according to the De Giorgi|2006}}|Complete list of De Giorgi's scientific publication (De Giorgi 2006, p. 6), an English translation should be included in {{harv|De Giorgi|2006}}, it is unfortunately missing.

}}. Translated in English as "On the differentiability and the analyticity of extremals of regular multiple integrals" in {{harv|De Giorgi|2006|pp=149–166}}.

}}. Translated in English as "An example of discontinuous extremals for a variational problem of elliptic type" in {{harv|De Giorgi|2006|pp=285–287}}.

}}), and in French (with additions of Hilbert himself) by M. L. Laugel as {{Citation