597:, 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
2913:
437:
1961:
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
84:
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
696:
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,
625:. 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
293:
110:, 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
685:
gave several counterexamples involving a single elliptic equation of order greater than two with analytic coefficients. For experts, the fact that such equations could have nonanalytic and even nonsmooth solutions created a sensation.
1315:
2823:
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,
3165:
280:
126:
as examples. 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:
3345:
2788:
1152:
657:
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
3191:
1770:
541:
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
918:
793:
1513:
2935:
1878:
1435:
3510:(1977), "Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity", in Kluge, Reinhard; Müller, Wolfdietrich (eds.),
525:" 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
2783:
2618:
2556:
432:{\displaystyle {\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}
3237:
1564:
1638:
1358:
2858:
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 (eds.),
1182:
1611:
988:
1584:
1052:
1032:
1012:
961:
941:
864:
836:
816:
3512:
Theory of nonlinear operators: constructive aspects. Proceedings of the fourth international summer school, held at Berlin, GDR, from
September 22 to 26, 1975
3507:
2249:
2132:... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt
593:
solutions of nonlinear elliptic analytic equations in 2 variables are analytic. Bernstein's result was improved over the years by several authors, such as
542:
1190:
523:... does every Lagrangian partial differential equation of a regular variation problem have the property of admitting analytic integrals exclusively?
3602:
3135:
3009:
2965:
139:
54:
with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of
2387:
2786:(1968), "Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni",
1946:, pp. 288–289), or the corresponding section on the nineteenth problem in any of its translations or reprints, or the subsection "
3066:
Compte Rendu du Deuxième Congrès
International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications
1064:
2665:, Classics in Mathematics (Revised 3rd printing of 2nd ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. xiv+517,
3309:, Die Grundlehren der mathematischen Wissenschaften, vol. 130, Berlin–Heidelberg–New York: Springer-Verlag, pp. xii+506,
103:
2820:(1999), "Vladimir Maz'ya: Friend and Mathematician. Recollections", in Rossman, Jürgen; Takáč, Peter; Wildenhain, Günther (eds.),
3765:
502:
51:
3314:
2908:
2879:
2832:
2753:
2670:
2626:
2582:
2399:
547:
3595:
3738:
3727:
3094:– There exists also an earlier (and shorter) resume of Hilbert's original talk, translated in French and published as
2473:
De Giorgi, Ennio (1957), "Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari",
3732:
3722:
2331:
1685:
3101:
3702:
3060:
2051:". Hilbert's definition of a regular variational problem is stronger than the one currently used, for example, in (
872:
3770:
3707:
3687:
3682:
3438:
678:, showing that in general there is no hope of proving such regularity results without adding further hypotheses.
3760:
3514:, Abhandlungen der Akademie der Wissenschaften der DDR, vol. 1, Berlin: Akademie-Verlag, pp. 197–206,
2457:" (English translation of the title) is a short research announcement disclosing the results detailed later in (
723:
3717:
3697:
3692:
3588:
3430:
2914:
Nachrichten von der Königlichen
Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
2510:
De Giorgi, Ennio (1968), "Un esempio di estremali discontinue per un problema variazionale di tipo ellittico",
1446:
1809:
1366:
3672:
3186:
3154:
2391:
1803:≥ 0. From his estimate Nash was able to deduce a continuity estimate for solutions of the elliptic equation
55:
3677:
3652:
62:
from the equation it satisfies. Hilbert's nineteenth problem was solved independently in the late 1950s by
2357:
3657:
3637:
3627:
3189:(1968), "Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients",
2955:
2930:
2273:
1922:
1667:+2)nd derivatives are also Hölder continuous, so repeating this infinitely often shows that the solution
1055:
567:
47:
2430:
Atti della
Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali
3667:
3662:
3647:
3642:
3632:
3622:
3535:
2708:
2560:
3078:
2277:
3142:
2827:, Operator Theory. Advances and Applications, vol. 109, Basel: Birkhäuser Verlag, pp. 1–5,
3251:
1525:
2475:
Memorie della
Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematicahe e Naturali
659:
506:
119:
3160:Примеры нерегулярных решений квазилинейных эллиптических уравнений с аналитическими коэффициентами
1566:
is bounded. When this is not the case, a further step is needed: one must prove that the solution
3544:
1616:
1323:
115:
1927:
Are the solutions of regular problems in the calculus of variations always necessarily analytic?
3300:
3246:
2737:
31:
2278:"Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre"
102:
David
Hilbert presented what is now called his nineteenth problem in his speech at the second
3611:
3141:(Report). Oxford: Oxford Centre for Nonlinear PDE. pp. 1–30. OxPDE-11/17. Archived from
2282:
1974:
111:
43:
3565:
3519:
3491:
3447:
3392:
3354:
3324:
3276:
3178:
3032:
2996:
2889:
2842:
2801:
2763:
2716:
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2636:
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2523:
2486:
2441:
2409:
2345:
1587:
1160:
67:
3573:
3557:
3527:
3499:
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3292:
3220:
3122:
3074:
3048:
2988:
2948:
2922:
2897:
2850:
2809:
2771:
2724:
2688:
2644:
2600:
2531:
2494:
2449:
2417:
2353:
2337:
Proceedings of the International Congress of Mathematicians, Vancouver, B.C., 1974, Vol. 1
2311:
622:
8:
2108:
3451:
3358:
1593:
970:
521:. Having identified the class of problems considered, he poses the following question: "
3479:
3380:
3280:
3208:
3036:
2500:
On the differentiability and the analyticity of extremals of regular multiple integrals
2315:
1991:
1569:
1037:
1034:
that satisfies certain growth, smoothness, and convexity conditions. The smoothness of
1017:
997:
946:
926:
849:
821:
801:
538:
58:, any solution inherits the relatively simple and well understood property of being an
3401:
2621:, vol. 105, Princeton, New Jersey: Princeton University Press, pp. vii+297,
3471:
3406:
3372:
3310:
3233:"Regularity of minima: an invitation to the Dark Side of the Calculus of Variations."
3228:
3212:
3131:
2875:
2828:
2749:
2666:
2622:
2608:
2578:
2395:
2319:
2299:
2073:
710:
59:
35:
3284:
3040:
2979:
3569:
3553:
3523:
3495:
3463:
3455:
3426:
3414:
3396:
3362:
3340:
3328:
3288:
3264:
3256:
3216:
3200:
3118:
3110:
3070:
3044:
3018:
2984:
2974:
2944:
2918:
2893:
2867:
2846:
2805:
2767:
2741:
2720:
2684:
2640:
2596:
2570:
2527:
2490:
2445:
2413:
2349:
2307:
2291:
1987:
1310:{\displaystyle \sum \limits _{i=1}^{n}(L_{p_{i}p_{j}}(Dw)w_{x_{j}x_{k}})_{x_{i}}=0}
638:
610:
526:
23:
3136:
Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas
3023:
717:
is a solution of a suitable linear second order strictly elliptic PDE of the form
3561:
3515:
3487:
3388:
3320:
3304:
3272:
3174:
3028:
2992:
2885:
2838:
2821:
2797:
2759:
2731:
2712:
2676:
2660:
2632:
2612:
2588:
2566:
2552:
2519:
2482:
2437:
2425:
2405:
2381:
2373:
2341:
2327:
2069:
626:
598:
63:
2871:
2537:
An example of discontinuous extremals for a variational problem of elliptic type
123:
3346:
Proceedings of the National Academy of Sciences of the United States of America
2817:
2656:
2614:
Multiple integrals in the calculus of variations and nonlinear elliptic systems
2548:
675:
3580:
3467:
3268:
3260:
3054:– Translated to French by M. L. Laugel (with additions of Hilbert himself) as
2574:
3754:
3475:
3376:
3096:
3056:
3004:
2960:
2904:
2779:
2696:
2652:
2377:
2340:, ICM Proceedings, Montreal: Canadian Mathematical Congress, pp. 53–63,
2303:
1679:
John Nash gave a continuity estimate for solutions of the parabolic equation
90:
27:
652:
3410:
3540:"Sur l'analyticité des solutions des systèmes d'équations différentielles"
3367:
3158:
2745:
2335:
1919:
Sind die Lösungen regulärer Variationsprobleme stets notwendig analytisch?
841:
2135:
3064:
3483:
3204:
3114:
2295:
2065:
621:), who were able to show the solutions had first derivatives that were
38:. Informally, and perhaps less directly, since Hilbert's concept of a "
2740:– London – Singapore: World Scientific Publishing, pp. viii+403,
701:
provided other, more refined, examples for the vector valued problem.
3384:
2428:(1956), "Sull'analiticità delle estremali degli integrali multipli",
275:{\displaystyle {\iint F(p,q,z;x,y)dxdy}={\text{Minimum}}\qquad \left}
3539:
3459:
2463:
Complete list of De Giorgi's scientific publication (De Giorgi 2006
991:
30:
in 1900. It asks whether the solutions of regular problems in the
2704:
492:
1147:{\displaystyle \sum \limits _{i=1}^{n}(L_{p_{i}}(Dw))_{x_{i}}=0}
122:
and a class of linear partial differential equations studied by
2016:
Unlike Liouville's work, Picard's work is explicitly cited by
3431:"Continuity of solutions of parabolic and elliptic equations"
3232:
3069:, ICM Proceedings, Paris: Gauthier-Villars, pp. 58–114,
2863:
2862:, Operator Theory: Advances and Applications, vol. 109,
1522:
has Hölder continuous first derivatives, provided the matrix
2565:, Springer Collected Works in Mathematics, Berlin–New York:
2465:, p. 6), an English translation should be included in (
1973:
For a detailed historical analysis, see the relevant entry "
1663:
th derivatives, so a theorem of Schauder implies that the (
2376:(1976), "Variational problems and elliptic equations", in
2076:, sense, even before the statement of its analyticity in
653:
Counterexamples to various generalizations of the problem
16:
When are solutions in the calculus of variations analytic
1058:
for this variational problem is the non-linear equation
566:
solutions, Hilbert's problem was answered positively by
505:
on the Euler–Lagrange equations associated to the given
2662:
Elliptic partial differential equations of second order
2383:
Mathematical developments arising from Hilbert problems
1054:
can be shown using De Giorgi's theorem as follows. The
842:
Application of De Giorgi's theorem to Hilbert's problem
2455:
On the analyticity of extremals of multiple integrals
1959:
English translation by Mary Frances Winston Newson:-"
1812:
1688:
1619:
1596:
1572:
1528:
1449:
1369:
1326:
1193:
1163:
1067:
1040:
1020:
1000:
973:
949:
929:
875:
852:
824:
804:
726:
515:
is a simple regularity assumption about the function
296:
142:
2126:
English translation by Mary Frances Winston Newson:
1014:
is the Lagrangian, a function of the derivatives of
532:
1929:"), formulating the problem with the same words of
3129:
2253:
2202:
2164:
1872:
1764:
1632:
1605:
1578:
1558:
1507:
1429:
1352:
1309:
1176:
1146:
1046:
1026:
1006:
982:
955:
935:
912:
858:
830:
810:
787:
431:
274:
3752:
3306:Multiple integrals in the calculus of variations
2651:
2052:
3610:
1947:
1765:{\displaystyle D_{i}(a^{ij}(x)D_{j}u)=D_{t}(u)}
3545:Recueil Mathématique (Matematicheskii Sbornik)
2390:, vol. XXVIII, Providence, Rhode Island:
846:Hilbert's problem asks whether the minimizers
818:has square integrable first derivatives, then
3596:
3010:Bulletin of the American Mathematical Society
2966:Bulletin of the American Mathematical Society
2778:
2332:"Variational problems and elliptic equations"
2030:
2028:
2026:
913:{\displaystyle \int _{U}L(Dw)\,\mathrm {d} x}
693:
671:
453:is an analytic function of all its arguments
78:
3061:"Sur les problèmes futurs des Mathématiques"
2733:Direct Methods in the Calculus of Variations
1908:) or, equivalently, one of its translations.
2701:Metodi diretti nel calcolo delle variazioni
2388:Proceedings of Symposia in Pure Mathematics
545:for equations belonging to this class. For
537:Hilbert stated his nineteenth problem as a
3603:
3589:
3153:
2789:Bollettino dell'Unione Matematica Italiana
2512:Bollettino dell'Unione Matematica Italiana
2023:
2020:, p. 288 and footnote 1 in the same page).
1950:" in the historical section of this entry.
788:{\displaystyle D_{i}(a^{ij}(x)\,D_{j}u)=0}
709:The key theorem proved by De Giorgi is an
682:
663:
3534:
3400:
3366:
3250:
3166:Funktsional'nyĭ Analiz I Ego Prilozheniya
3022:
2978:
2607:
2546:
2540:
2509:
2503:
2472:
2466:
2462:
2458:
2424:
2272:
2232:
2181:
2148:
1508:{\displaystyle a^{ij}=L_{p_{i}p_{j}}(Dw)}
1157:and differentiating this with respect to
901:
762:
689:
667:
634:
630:
606:
602:
594:
571:
3227:
3192:Functional Analysis and Its Applications
2372:
2326:
2257:
2206:
2168:
1873:{\displaystyle D_{i}(a^{ij}(x)D_{j}u)=0}
1430:{\displaystyle D_{i}(a^{ij}(x)D_{j}u)=0}
104:International Congress of Mathematicians
2903:
2857:
2816:
2703:, Monografie Matematiche (in Italian),
2248:For more information about the work of
2219:
2198:
2160:
2156:
2127:
2035:
2017:
2004:
1943:
1930:
1905:
107:
94:
3753:
3299:
2729:
2695:
2236:
2185:
2152:
1518:so by De Giorgi's result the solution
704:
52:elliptic partial differential equation
3584:
3506:
2239:, p. 7, pp. 202–203 and pp. 317–318).
1880:by considering the special case when
698:
3425:
3339:
2866:: Birkhäuser Verlag, pp. 7–16,
2068:in the "classical", i.e. not in the
1647:is known to have Hölder continuous (
646:
642:
618:
614:
3099:(1900), "Problèmes mathématiques",
2003:: compare the relevant entry with (
1195:
1069:
13:
1625:
943:is a function on some compact set
903:
674:independently constructed several
403:
395:
380:
353:
338:
316:
301:
252:
244:
221:
213:
14:
3782:
3007:(2000), "Mathematical Problems",
2963:(1902), "Mathematical Problems",
2860:The Maz'ya Anniversary Collection
1986:Hilbert does not cite explicitly
1674:
533:The path to the complete solution
85:analytischer Lösungen fähig sind.
42:" identifies this precisely as a
2936:Archiv der Mathematik und Physik
2155:, p. 7 footnote 7 and p. 353), (
866:of an energy functional such as
574:) in his thesis. He showed that
26:, set out in a list compiled by
3439:American Journal of Mathematics
3343:(1957), "Parabolic equations",
2980:10.1090/S0002-9904-1902-00923-3
2469:), it is unfortunately missing.
2254:Kristensen & Mingione (2011
2242:
2225:
2212:
2191:
2174:
2141:
2120:
2058:
240:
236:
204:
3766:Partial differential equations
2203:Kristensen & Mingione 2011
2165:Kristensen & Mingione 2011
2130:, p. 288) precise words are:-"
2041:
2010:
1980:
1967:
1953:
1936:
1911:
1898:
1861:
1845:
1839:
1823:
1759:
1753:
1737:
1721:
1715:
1699:
1559:{\displaystyle L_{p_{i}p_{j}}}
1502:
1493:
1418:
1402:
1396:
1380:
1360:satisfies the linear equation
1285:
1254:
1245:
1215:
1122:
1118:
1109:
1089:
898:
889:
776:
759:
753:
737:
445:
288:
180:
150:
134:
56:partial differential equations
1:
3024:10.1090/S0273-0979-00-00881-8
2619:Annals of Mathematics Studies
2392:American Mathematical Society
2266:
2113:
2078:
511:
497:
487:
2939:, dritte reihe (in German),
2535:. Translated in English as "
2498:. Translated in English as "
2064:Since Hilbert considers all
2053:Gilbarg & Trudinger 2001
20:Hilbert's nineteenth problem
7:
3238:Applications of Mathematics
3184:– Translated in English as
3102:L'Enseignement Mathématique
2956:Mary Frances Winston Newson
2954:– Translated to English by
2943:: 44–63 and 253–297, 1900,
2872:10.1007/978-3-0348-8675-8_2
2728:, translated in English as
2559:; Spagnolo, Sergio (eds.),
2461:). While, according to the
2432:, Serie VIII (in Italian),
2049:Reguläres Variationsproblem
1990:and considers the constant
1923:Mary Frances Winston Newson
1655:≥ 1, then the coefficients
1651:+1)st derivatives for some
1633:{\displaystyle L^{\infty }}
1353:{\displaystyle u=w_{x_{k}}}
694:Giusti & Miranda (1968)
672:Giusti & Miranda (1968)
662:. At the end of the 1960s,
483:regular variational problem
442:
285:
131:
40:regular variational problem
10:
3787:
2917:(in German) (3): 253–297,
2709:Unione Matematica Italiana
2477:, Serie III (in Italian),
2088:is assumed to be at least
1948:The origins of the problem
1921:" (English translation by
79:The origins of the problem
73:
3618:
3261:10.1007/s10778-006-0110-3
2792:, Serie IV (in Italian),
2575:10.1007/978-3-642-41496-1
2547:De Giorgi, Ennio (2006),
2514:, Serie IV (in Italian),
1779:is a bounded function of
3159:
3063:, in Duporcq, E. (ed.),
2931:"Mathematische Probleme"
2909:"Mathematische Probleme"
1892:
120:minimal surface equation
2730:Giusti, Enrico (2003),
2256:, §3.3, pp. 9–12) and (
2235:, pp. 54–59) and (
2167:, p. 5 and p. 8), and (
1659:have Hölder continuous
1056:Euler–Lagrange equation
48:Euler–Lagrange equation
3771:Calculus of variations
2738:River Edge, New Jersey
2205:, p. 5 and p. 8) and (
1874:
1766:
1634:
1607:
1580:
1560:
1509:
1431:
1354:
1311:
1214:
1178:
1148:
1088:
1048:
1028:
1008:
984:
957:
937:
914:
860:
838:is Hölder continuous.
832:
812:
789:
481:Hilbert calls this a "
433:
276:
100:
32:calculus of variations
3368:10.1073/pnas.43.8.754
2746:10.1142/9789812795557
2283:Mathematische Annalen
2260:, §3.3, pp. 369–370).
2047:In his exact words: "
1875:
1767:
1635:
1608:
1581:
1561:
1510:
1432:
1355:
1312:
1194:
1179:
1177:{\displaystyle x_{k}}
1149:
1068:
1049:
1029:
1009:
985:
958:
938:
915:
861:
833:
813:
790:
503:ellipticity condition
491:means that these are
434:
277:
82:
2543:, pp. 285–287).
2506:, pp. 149–166).
2394:, pp. 525–535,
2201:, pp. 10–11), (
2188:, p. 7 and pp. 353).
2184:, pp. 54–59), (
2163:, pp. 10–11), (
2138:by Hilbert himself).
2107:, as the use of the
1810:
1686:
1617:
1594:
1590:, i.e. the gradient
1588:Lipschitz continuous
1570:
1526:
1447:
1367:
1324:
1191:
1161:
1065:
1038:
1018:
998:
971:
947:
927:
873:
850:
822:
802:
724:
639:John Forbes Nash
611:John Forbes Nash
568:Sergei Bernstein
294:
140:
116:Liouville's equation
68:John Forbes Nash, Jr
3452:1958AmJM...80..931N
3359:1957PNAS...43..754N
2711:, pp. VI+422,
2109:Hessian determinant
1884:does not depend on
923:are analytic. Here
705:De Giorgi's theorem
627:Ennio De Giorgi
599:Ennio De Giorgi
543:classical solutions
44:variational problem
3761:Hilbert's problems
3612:Hilbert's problems
3468:10338.dmlcz/101876
3301:Morrey, Charles B.
3269:10338.dmlcz/134645
3229:Mingione, Giuseppe
3205:10.1007/BF01076124
3132:Mingione, Giuseppe
3115:10.5169/seals-3575
2657:Trudinger, Neil S.
2609:Giaquinta, Mariano
2569:, pp. x+889,
2296:10.1007/BF01444746
1992:Gaussian curvature
1975:Hilbert's problems
1962:analytic solutions
1870:
1762:
1630:
1606:{\displaystyle Dw}
1603:
1576:
1556:
1505:
1427:
1350:
1307:
1174:
1144:
1044:
1024:
1004:
983:{\displaystyle Dw}
980:
953:
933:
910:
856:
828:
808:
785:
539:regularity problem
527:potential function
429:
272:
112:Laplace's equation
3748:
3747:
3316:978-3-540-69915-6
3130:Kristensen, Jan;
2881:978-3-0348-9726-6
2834:978-3-7643-6201-0
2755:978-981-238-043-2
2672:978-3-540-41160-4
2628:978-0-691-08330-8
2584:978-3-540-26169-8
2401:978-0-8218-1428-4
2378:Browder, Felix E.
1579:{\displaystyle w}
1047:{\displaystyle w}
1027:{\displaystyle w}
1007:{\displaystyle L}
956:{\displaystyle U}
936:{\displaystyle w}
859:{\displaystyle w}
831:{\displaystyle u}
811:{\displaystyle u}
711:a priori estimate
623:Hölder continuous
509:, while property
411:
366:
329:
259:
228:
202:
60:analytic function
22:is one of the 23
3778:
3605:
3598:
3591:
3582:
3581:
3576:
3536:Petrowsky, I. G.
3530:
3502:
3435:
3421:
3404:
3370:
3335:
3295:
3254:
3223:
3181:
3149:
3147:
3140:
3134:(October 2011).
3125:
3091:
3090:
3089:
3083:
3077:, archived from
3051:
3026:
2999:
2982:
2951:
2925:
2900:
2853:
2812:
2774:
2727:
2691:
2647:
2603:
2555:; Forti, Marco;
2553:Dal Maso, Gianni
2534:
2497:
2452:
2426:De Giorgi, Ennio
2420:
2374:Bombieri, Enrico
2370:
2369:
2368:
2362:
2356:, archived from
2328:Bombieri, Enrico
2322:
2261:
2252:see the work of
2246:
2240:
2229:
2223:
2216:
2210:
2195:
2189:
2178:
2172:
2151:, p. 59), (
2145:
2139:
2136:Italics emphasis
2124:
2118:
2106:
2105:
2104:
2103:
2100:
2096:
2093:
2087:
2062:
2056:
2045:
2039:
2032:
2021:
2014:
2008:
2002:
1998:
1988:Joseph Liouville
1984:
1978:
1971:
1965:
1957:
1951:
1940:
1934:
1915:
1909:
1902:
1879:
1877:
1876:
1871:
1857:
1856:
1838:
1837:
1822:
1821:
1771:
1769:
1768:
1763:
1752:
1751:
1733:
1732:
1714:
1713:
1698:
1697:
1639:
1637:
1636:
1631:
1629:
1628:
1612:
1610:
1609:
1604:
1585:
1583:
1582:
1577:
1565:
1563:
1562:
1557:
1555:
1554:
1553:
1552:
1543:
1542:
1514:
1512:
1511:
1506:
1492:
1491:
1490:
1489:
1480:
1479:
1462:
1461:
1436:
1434:
1433:
1428:
1414:
1413:
1395:
1394:
1379:
1378:
1359:
1357:
1356:
1351:
1349:
1348:
1347:
1346:
1320:This means that
1316:
1314:
1313:
1308:
1300:
1299:
1298:
1297:
1283:
1282:
1281:
1280:
1271:
1270:
1244:
1243:
1242:
1241:
1232:
1231:
1213:
1208:
1183:
1181:
1180:
1175:
1173:
1172:
1153:
1151:
1150:
1145:
1137:
1136:
1135:
1134:
1108:
1107:
1106:
1105:
1087:
1082:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1013:
1011:
1010:
1005:
989:
987:
986:
981:
962:
960:
959:
954:
942:
940:
939:
934:
919:
917:
916:
911:
906:
885:
884:
865:
863:
862:
857:
837:
835:
834:
829:
817:
815:
814:
809:
794:
792:
791:
786:
772:
771:
752:
751:
736:
735:
713:stating that if
690:De Giorgi (1968)
668:De Giorgi (1968)
595:Petrowsky (1939)
592:
591:
590:
589:
586:
582:
579:
565:
563:
562:
561:
558:
554:
551:
520:
493:minimum problems
476:
470:
452:
447:
446:
438:
436:
435:
430:
422:
421:
416:
412:
410:
409:
401:
392:
388:
387:
377:
367:
365:
361:
360:
350:
346:
345:
335:
330:
328:
324:
323:
313:
309:
308:
298:
290:
289:
281:
279:
278:
273:
271:
267:
260:
258:
250:
242:
229:
227:
219:
211:
203:
200:
195:
136:
135:
98:
24:Hilbert problems
3786:
3785:
3781:
3780:
3779:
3777:
3776:
3775:
3751:
3750:
3749:
3744:
3614:
3609:
3508:Nečas, Jindřich
3460:10.2307/2372841
3433:
3317:
3252:10.1.1.214.9183
3185:
3183:
3161:
3145:
3138:
3095:
3093:
3087:
3085:
3081:
3055:
3053:
3003:
3002:– Reprinted as
3001:
2973:(10): 437–479,
2959:
2953:
2929:
2928:– Reprinted as
2927:
2882:
2835:
2818:Gohberg, Israel
2756:
2673:
2629:
2585:
2567:Springer-Verlag
2562:Selected papers
2549:Ambrosio, Luigi
2402:
2371:. Reprinted in
2366:
2364:
2360:
2269:
2264:
2247:
2243:
2230:
2226:
2217:
2213:
2209:, p. 368).
2196:
2192:
2179:
2175:
2171:, p. 368).
2159:, p. 1), (
2146:
2142:
2128:Hilbert's (1900
2125:
2121:
2101:
2098:
2097:
2094:
2091:
2090:
2089:
2083:
2082:, the function
2063:
2059:
2055:, p. 289).
2046:
2042:
2038:, p. 288).
2033:
2024:
2015:
2011:
2007:, p. 288).
2000:
1994:
1985:
1981:
1972:
1968:
1958:
1954:
1941:
1937:
1933:, p. 288).
1916:
1912:
1903:
1899:
1895:
1852:
1848:
1830:
1826:
1817:
1813:
1811:
1808:
1807:
1794:
1785:
1747:
1743:
1728:
1724:
1706:
1702:
1693:
1689:
1687:
1684:
1683:
1677:
1624:
1620:
1618:
1615:
1614:
1595:
1592:
1591:
1571:
1568:
1567:
1548:
1544:
1538:
1534:
1533:
1529:
1527:
1524:
1523:
1485:
1481:
1475:
1471:
1470:
1466:
1454:
1450:
1448:
1445:
1444:
1409:
1405:
1387:
1383:
1374:
1370:
1368:
1365:
1364:
1342:
1338:
1337:
1333:
1325:
1322:
1321:
1293:
1289:
1288:
1284:
1276:
1272:
1266:
1262:
1261:
1257:
1237:
1233:
1227:
1223:
1222:
1218:
1209:
1198:
1192:
1189:
1188:
1168:
1164:
1162:
1159:
1158:
1130:
1126:
1125:
1121:
1101:
1097:
1096:
1092:
1083:
1072:
1066:
1063:
1062:
1039:
1036:
1035:
1019:
1016:
1015:
999:
996:
995:
972:
969:
968:
948:
945:
944:
928:
925:
924:
902:
880:
876:
874:
871:
870:
851:
848:
847:
844:
823:
820:
819:
803:
800:
799:
767:
763:
744:
740:
731:
727:
725:
722:
721:
707:
676:counterexamples
655:
587:
584:
583:
580:
577:
576:
575:
559:
556:
555:
552:
549:
548:
546:
535:
516:
472:
454:
448:
444:
417:
402:
394:
393:
383:
379:
378:
376:
372:
371:
356:
352:
351:
341:
337:
336:
334:
319:
315:
314:
304:
300:
299:
297:
295:
292:
291:
287:
251:
243:
241:
220:
212:
210:
209:
205:
199:
143:
141:
138:
137:
133:
99:
97:, p. 288).
89:
81:
76:
64:Ennio De Giorgi
17:
12:
11:
5:
3784:
3774:
3773:
3768:
3763:
3746:
3745:
3743:
3742:
3735:
3730:
3725:
3720:
3715:
3710:
3705:
3700:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3660:
3655:
3650:
3645:
3640:
3635:
3630:
3625:
3619:
3616:
3615:
3608:
3607:
3600:
3593:
3585:
3579:
3578:
3532:
3504:
3446:(4): 931–954,
3423:
3353:(8): 754–758,
3337:
3315:
3297:
3245:(4): 355–426,
3225:
3199:(3): 230–234,
3169:(in Russian),
3151:
3148:on 2014-01-07.
3127:
3057:Hilbert, David
3017:(4): 407–436,
3013:, New Series,
3005:Hilbert, David
2961:Hilbert, David
2905:Hilbert, David
2901:
2880:
2855:
2833:
2814:
2784:Miranda, Mario
2780:Giusti, Enrico
2776:
2754:
2697:Giusti, Enrico
2693:
2671:
2653:Gilbarg, David
2649:
2627:
2605:
2583:
2557:Miranda, Mario
2544:
2541:De Giorgi 2006
2507:
2504:De Giorgi 2006
2470:
2467:De Giorgi 2006
2459:De Giorgi 1957
2422:
2400:
2324:
2290:(1–2): 20–76,
2268:
2265:
2263:
2262:
2250:Jindřich Nečas
2241:
2233:Giaquinta 1983
2224:
2218:According to (
2211:
2190:
2182:Giaquinta 1983
2173:
2149:Giaquinta 1983
2140:
2119:
2057:
2040:
2022:
2009:
1979:
1966:
1952:
1935:
1910:
1896:
1894:
1891:
1890:
1889:
1869:
1866:
1863:
1860:
1855:
1851:
1847:
1844:
1841:
1836:
1833:
1829:
1825:
1820:
1816:
1790:
1783:
1773:
1772:
1761:
1758:
1755:
1750:
1746:
1742:
1739:
1736:
1731:
1727:
1723:
1720:
1717:
1712:
1709:
1705:
1701:
1696:
1692:
1676:
1675:Nash's theorem
1673:
1627:
1623:
1602:
1599:
1575:
1551:
1547:
1541:
1537:
1532:
1516:
1515:
1504:
1501:
1498:
1495:
1488:
1484:
1478:
1474:
1469:
1465:
1460:
1457:
1453:
1438:
1437:
1426:
1423:
1420:
1417:
1412:
1408:
1404:
1401:
1398:
1393:
1390:
1386:
1382:
1377:
1373:
1345:
1341:
1336:
1332:
1329:
1318:
1317:
1306:
1303:
1296:
1292:
1287:
1279:
1275:
1269:
1265:
1260:
1256:
1253:
1250:
1247:
1240:
1236:
1230:
1226:
1221:
1217:
1212:
1207:
1204:
1201:
1197:
1171:
1167:
1155:
1154:
1143:
1140:
1133:
1129:
1124:
1120:
1117:
1114:
1111:
1104:
1100:
1095:
1091:
1086:
1081:
1078:
1075:
1071:
1043:
1023:
1003:
979:
976:
952:
932:
921:
920:
909:
905:
900:
897:
894:
891:
888:
883:
879:
855:
843:
840:
827:
807:
796:
795:
784:
781:
778:
775:
770:
766:
761:
758:
755:
750:
747:
743:
739:
734:
730:
706:
703:
654:
651:
534:
531:
479:
478:
440:
428:
425:
420:
415:
408:
405:
400:
397:
391:
386:
382:
375:
370:
364:
359:
355:
349:
344:
340:
333:
327:
322:
318:
312:
307:
303:
283:
270:
266:
263:
257:
254:
249:
246:
239:
235:
232:
226:
223:
218:
215:
208:
198:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
87:
80:
77:
75:
72:
15:
9:
6:
4:
3:
2:
3783:
3772:
3769:
3767:
3764:
3762:
3759:
3758:
3756:
3740:
3736:
3734:
3731:
3729:
3726:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3620:
3617:
3613:
3606:
3601:
3599:
3594:
3592:
3587:
3586:
3583:
3575:
3571:
3567:
3563:
3559:
3555:
3551:
3548:(in French),
3547:
3546:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3497:
3493:
3489:
3485:
3481:
3477:
3473:
3469:
3465:
3461:
3457:
3453:
3449:
3445:
3441:
3440:
3432:
3428:
3424:
3420:
3416:
3412:
3408:
3403:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3369:
3364:
3360:
3356:
3352:
3348:
3347:
3342:
3338:
3334:
3330:
3326:
3322:
3318:
3312:
3308:
3307:
3302:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3270:
3266:
3262:
3258:
3253:
3248:
3244:
3240:
3239:
3234:
3230:
3226:
3222:
3218:
3214:
3210:
3206:
3202:
3198:
3194:
3193:
3188:
3187:Maz'ya, V. G.
3180:
3176:
3172:
3168:
3167:
3162:
3156:
3155:Maz'ya, V. G.
3152:
3144:
3137:
3133:
3128:
3124:
3120:
3116:
3112:
3108:
3105:(in French),
3104:
3103:
3098:
3084:on 2013-12-31
3080:
3076:
3072:
3068:
3067:
3062:
3058:
3050:
3046:
3042:
3038:
3034:
3030:
3025:
3020:
3016:
3012:
3011:
3006:
2998:
2994:
2990:
2986:
2981:
2976:
2972:
2968:
2967:
2962:
2957:
2950:
2946:
2942:
2938:
2937:
2932:
2924:
2920:
2916:
2915:
2910:
2906:
2902:
2899:
2895:
2891:
2887:
2883:
2877:
2873:
2869:
2865:
2861:
2856:
2852:
2848:
2844:
2840:
2836:
2830:
2826:
2825:
2819:
2815:
2811:
2807:
2803:
2799:
2795:
2791:
2790:
2785:
2781:
2777:
2773:
2769:
2765:
2761:
2757:
2751:
2747:
2743:
2739:
2735:
2734:
2726:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2668:
2664:
2663:
2658:
2654:
2650:
2646:
2642:
2638:
2634:
2630:
2624:
2620:
2616:
2615:
2610:
2606:
2602:
2598:
2594:
2590:
2586:
2580:
2576:
2572:
2568:
2564:
2563:
2558:
2554:
2550:
2545:
2542:
2538:
2533:
2529:
2525:
2521:
2517:
2513:
2508:
2505:
2501:
2496:
2492:
2488:
2484:
2480:
2476:
2471:
2468:
2464:
2460:
2456:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2397:
2393:
2389:
2385:
2384:
2379:
2375:
2363:on 2013-12-31
2359:
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2297:
2293:
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2286:(in French),
2285:
2284:
2279:
2275:
2274:Bernstein, S.
2271:
2270:
2259:
2258:Mingione 2006
2255:
2251:
2245:
2238:
2234:
2228:
2222:, p. 1).
2221:
2215:
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2207:Mingione 2006
2204:
2200:
2194:
2187:
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2169:Mingione 2006
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2061:
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2029:
2027:
2019:
2018:Hilbert (1900
2013:
2006:
1997:
1993:
1989:
1983:
1976:
1970:
1963:
1956:
1949:
1945:
1939:
1932:
1931:Hilbert (1900
1928:
1924:
1920:
1914:
1907:
1901:
1897:
1887:
1883:
1867:
1864:
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993:
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741:
732:
728:
720:
719:
718:
716:
712:
702:
700:
695:
691:
687:
684:
683:Maz'ya (1968)
679:
677:
673:
669:
665:
664:Maz'ya (1968)
661:
650:
648:
644:
640:
636:
632:
628:
624:
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604:
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255:
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206:
196:
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177:
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159:
156:
153:
147:
144:
130:
129:
128:
125:
121:
117:
113:
109:
105:
96:
92:
91:David Hilbert
86:
71:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
28:David Hilbert
25:
21:
3712:
3552:(47): 3–70,
3549:
3543:
3511:
3443:
3437:
3350:
3344:
3305:
3242:
3236:
3196:
3190:
3173:(3): 53–57,
3170:
3164:
3143:the original
3106:
3100:
3086:, retrieved
3079:the original
3065:
3014:
3008:
2970:
2964:
2940:
2934:
2912:
2859:
2822:
2793:
2787:
2732:
2700:
2661:
2613:
2561:
2536:
2515:
2511:
2499:
2478:
2474:
2454:
2433:
2429:
2382:
2365:, retrieved
2358:the original
2336:
2287:
2281:
2244:
2227:
2220:Gohberg 1999
2214:
2199:Hedberg 1999
2193:
2176:
2161:Hedberg 1999
2157:Gohberg 1999
2143:
2131:
2122:
2112:
2084:
2077:
2060:
2048:
2043:
2036:Hilbert 1900
2012:
2005:Hilbert 1900
1999:as equal to
1995:
1982:
1969:
1960:
1955:
1944:Hilbert 1900
1938:
1926:
1918:
1913:
1906:Hilbert 1900
1900:
1885:
1881:
1800:
1799:defined for
1796:
1791:
1787:
1780:
1776:
1774:
1678:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1642:
1519:
1517:
1439:
1319:
1156:
994:vector, and
964:
922:
845:
797:
714:
708:
699:Nečas (1977)
688:
680:
656:
536:
522:
517:
510:
496:
486:
485:". Property
482:
480:
473:
467:
463:
459:
455:
449:
124:Émile Picard
108:Hilbert 1900
101:
95:Hilbert 1900
83:
39:
19:
18:
3109:: 349–355,
3097:Hilbert, D.
2518:: 135–137,
2436:: 438–441,
2237:Giusti 1994
2186:Giusti 1994
2153:Giusti 1994
2072:but in the
2066:derivatives
1671:is smooth.
681:Precisely,
660:functionals
495:. Property
34:are always
3755:Categories
3574:0022.22601
3558:65.0405.02
3528:0372.35031
3500:0096.06902
3427:Nash, John
3419:0078.08704
3341:Nash, John
3333:0142.38701
3293:1164.49324
3221:0179.43601
3123:31.0905.03
3088:2013-12-28
3075:32.0084.06
3049:0979.01028
2989:33.0976.07
2949:32.0084.05
2923:31.0068.03
2898:0939.31001
2851:0939.01018
2810:0155.44501
2772:1028.49001
2725:0942.49002
2689:1042.35002
2645:0516.49003
2601:1096.01015
2532:0084.31901
2495:0084.31901
2450:0074.31503
2418:0347.35032
2367:2011-01-29
2354:0344.49002
2312:35.0354.01
2267:References
1640:function.
507:functional
3476:0002-9327
3377:0027-8424
3247:CiteSeerX
3213:121038871
2659:(2001) ,
2481:: 25–43,
2320:121487650
2304:0025-5831
1626:∞
1196:∑
1070:∑
878:∫
404:∂
396:∂
381:∂
369:−
354:∂
339:∂
332:⋅
317:∂
302:∂
253:∂
245:∂
222:∂
214:∂
145:∬
3538:(1939),
3429:(1958),
3411:16590082
3303:(1966),
3285:16385131
3231:(2006),
3157:(1968),
3059:(1902),
3041:12695502
2907:(1900),
2699:(1994),
2611:(1983),
2330:(1975),
2276:(1904),
2117:implies.
992:gradient
88:—
36:analytic
3566:0001425
3520:0509483
3492:0100158
3484:2372841
3448:Bibcode
3393:0089986
3355:Bibcode
3325:0202511
3277:2291779
3179:0237946
3033:1779412
2997:1557926
2890:1747862
2843:1747861
2802:0232265
2796:: 1–8,
2764:1962933
2717:1707291
2705:Bologna
2681:1814364
2637:0717034
2593:2229237
2524:0227827
2487:0093649
2442:0082045
2410:0425740
2380:(ed.),
2346:0509259
990:is its
641: (
637:), and
629: (
613: (
609:), and
601: (
570: (
501:is the
201:Minimum
74:History
3572:
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3482:
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3402:528534
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2074:strong
1775:where
1613:is an
1440:with
1184:gives
588:
581:
560:
553:
118:, the
106:. In (
50:is an
46:whose
3480:JSTOR
3434:(PDF)
3385:89599
3381:JSTOR
3281:S2CID
3209:S2CID
3146:(PDF)
3139:(PDF)
3082:(PDF)
3037:S2CID
2864:Basel
2361:(PDF)
2316:S2CID
2231:See (
2197:See (
2180:See (
2147:See (
2034:See (
1942:See (
1904:See (
1893:Notes
1786:,...,
1643:Once
3472:ISSN
3407:PMID
3373:ISSN
3311:ISBN
2876:ISBN
2829:ISBN
2824:1998
2750:ISBN
2667:ISBN
2623:ISBN
2579:ISBN
2396:ISBN
2300:ISSN
2070:weak
2001:-1/2
798:and
692:and
670:and
647:1958
643:1957
635:1957
631:1956
619:1958
615:1957
607:1957
603:1956
572:1904
471:and
424:>
66:and
3570:Zbl
3554:JFM
3524:Zbl
3496:Zbl
3464:hdl
3456:doi
3415:Zbl
3397:PMC
3363:doi
3329:Zbl
3289:Zbl
3265:hdl
3257:doi
3217:Zbl
3201:doi
3119:JFM
3111:doi
3071:JFM
3045:Zbl
3019:doi
2985:JFM
2975:doi
2958:as
2945:JFM
2919:JFM
2894:Zbl
2868:doi
2847:Zbl
2806:Zbl
2768:Zbl
2742:doi
2721:Zbl
2685:Zbl
2641:Zbl
2597:Zbl
2571:doi
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2491:Zbl
2453:. "
2446:Zbl
2414:Zbl
2350:Zbl
2308:JFM
2292:doi
2134:" (
2114:(2)
2111:in
2079:(3)
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1964:".
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