Multiple integral problems in the calculus of variations and related topics

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1. [1] N. Aronszajn and K.T. Smith, Functional spaces and functional completion, Ann. Inst. Fourier Grenoble6 (1956), 125-185. Zbl0071.33003MR80878
2. [2] F.E. Browder, Strongly elliptic systems of differential equations, Contributions to the theory of partial differential equations, 15-51. Ann. of Math. Studies, No 33.Princeton University Press (1954). Zbl0057.32901MR67306
3. [3] — , Numerous notes in Proc, Nat. Acad. Sci. U.S.A.38 (1952), 230-235 and 741-747; 39 (1953), 179-184 and 185-190; and many others. 
4. [4] J.W. Calkin, Functions of several variables and ab80lute continuity, I, Duke Math. J.6 (1940), 170-185. Zbl0026.39202MR1278JFM66.1224.03
5. [5] L. Cesari, An éxistence theorem of the calculics of variations for integrals on parametric surfaces, Amer. J. Math.74 (1952); 265-295. Zbl0046.10901MR50187
6. [6] S. Cinquini, Sopra l'estremo assoluto degli integrali doppi in forma ordinaria, Ann. Math. Pura Appl, (4) 30 (1949) 249-260. Zbl0041.06801MR36458
7. [7] R. Courant, Plateau's problem and Dirichlet's principle, Ann. of Math. (2) 38 (1937) 679-724. Zbl0017.26802MR1503362JFM63.1067.01
8. [8] — , Dirichlet's principle, conformal mapping, and minimal surfaces, Interscience Press, New York (1950). Zbl0040.34603
9. [9] J.M. Danskin, On the existence of minimizing surfaces in parametric double integral problems in the calculus of variations, Rev. Math. Univ. Parma3 (1952) 43-63. Zbl0048.08104MR50186
10. [10] E. De Giorgi, Sull'analiticità delle extremali degli integrali multipli, Atti Accad. Naz. dei Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 20 (1956), 438-441. Zbl0074.31503MR82045
11. [11] J. Deny, Les potentiels d'energie finiActa Math.82 (1950), 107-183. Zbl0034.36201MR36371
12. [12] G. De Rham and K. Kodaira, Harmonic Integrals (mimeographed notes), Institute for Advanced Study, Princeton, N. J. (1950). MR37549
13. [13] G.F.D. Duff and D.C. Spencer, Harmonic tensors on Riemannian manifolds with boundary, Ann. of Math.56 (1952), 128-156. Zbl0049.18901MR48137
14. [14] G.C. Evans, Fundamental points of potential theory, Rice Inst. Pamphlets7 (1920), 252-359. Zbl48.1268.01JFM48.1268.01
15. [15] — , Note on a theorem of Bôcher, Amer. J. Math.50 (1928), 123-126. MR1506657JFM54.0508.01
16. [16] - , Potentials of positive mass I. Trans. Amer. Math. Soc.37 (1935), 226-253. Zbl61.0529.03MR1501785JFM61.0529.03
17. [17] G Fichera, Esistenza del minimo in un ctassico problema en calcul delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Soi. Fis. Mat. Naz. (8) 11 (1951), 34-39. Zbl0054.04303MR46588
18. [18] K.O. Friedrich, On the identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc.55 (1944), 132-151. Zbl0061.26201MR9701
19. [19] - , On the differentiability of the solutions of linear elliptic differential equations, Comm. Pure Appl. Math.6 (1953), 299-326. Zbl0051.32703MR58828
20. [20] - , On differential forms on Riemannian manifolds, Comm. Pure Appl. Math.8 (1955), 551-558. Zbl0066.07504MR87763
21. [21] G. Fubini, Il principio di minimo e i teoremi di esistenza per i problemi di contorno relativi alle equazioni alle derivate parziali di ordine pari, Rend. Circ. Mat. Palermo23 (1907), 58-84. Zbl38.0390.02
22. [22] M.P. Caffney, The harmonic operator for exterior differential forms, Proc. Nat. Acad. Sci. U.S.A.37 (1951), 48-50. Zbl0042.10205MR48138
23. [23] — , The heat equation method of Milgram and Roeenbloom for open Riemanniam manifolds, Ann. Of Math.60 (1954), 458-466. Zbl0057.07501MR64933
24. [24] L. Garding, Dirichlet's problem for linear elliptic partial differential equations, Math. Scad.1 (1953), 55-72. Zbl0053.39101
25. [25] W.V.D. Hodge, A Dirichlet problem for harmonic functionals with applications to analytic varieties, Proc. London Math. Soc. (2) 36 (1935), 257-303. Zbl0008.02203JFM59.1091.01
26. [26] — , The Theory and Applications of Harmonic Integrals, Second Edition, Cambridge University Press1952. Zbl0048.15702MR51571
27. [27] E. Hopf, Zum analytischen Charakter der Lösungen regulärer zweidimensionaler Variationsprobleme, Math. Zeit.30 (1929), 404-413. Zbl55.0898.03MR1545070JFM55.0898.03
28. [28] F. John, Derivatives of continuous weak solutious of linear elliptic equations, Comm. Pure Appl. Math.6 (1953) 327-335. Zbl0051.32801MR58829
29. [29] K. Kodaira, Harmonic fields in Riemannian manifolda, Ann. of Math.50 (1949), 587-665. Zbl0034.20502MR31148
30. [30] P.D. Lax, On Cauchy's problem for hyperboLic equations and the differentiability of the solutions of elliptic equations, Comm. Pure Appl. Math.8 (1955), 615-633. Zbl0067.07502MR78558
31. [31] H. Lebesgue, Sur Le problème de Dirichlet, Rend. Circ. Mat. Palermo24 (1907), 371-402. JFM38.0392.01
32. [32] B. Lrvi, Sul principio di Dirichlet, Rend. Ciro. Mat. Palermo22 (1906). 293-359. JFM37.0414.06
33. [33] H. Lewy, On minimal surfaoes with partially free boundary, Comm. Pure Appl. Math.4 (1952), 1-13. 
34. [34] L. Lichtenstein, Uber den analytischen Charakter der Lösungen zweidimensionaler Variatiosprobleme, Bull. Acad. Sci. Cracovia, Cl. Sci. Mat. Nat. (A) (1912), 915-941. JFM43.0470.01
35. [35] — , Zur Theorie der Konforme Abbildutig. Konforme Abbildung nichtanalyticher singularitatenfreier FLachenstiiche auf ebene Gebiete, Bull. Int. Acad. Sci. Cracovia, Mat. Nat. Cl. (A) (1916), 192-217. 
36. [36] E.J. Mcshane, Integrals ovsr surfaces in parametric form, Ann. of Math.34 (1933), 815-838. Zbl0008.07201MR1503135JFM59.0502.01
37. [37] — , Parametrizations of saddle surfaces with application to the problem of Plateau, Trans. Amer. Math. Soc.35 (1934), 718-733. Zbl0007.11902JFM59.0502.02
38. [38] A. Milgram and P. Rosenbloom, Harmoníc forms and heat conduction, Proc. Nat. Acad. Soi. U.S.A.37 (1951), 180-184 and 435-438. Zbl0044.31703
39. [39] C.B. Morhey, Jr. On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soo.43 (1938), 126-166. Zbl0018.40501MR1501936JFM64.0460.02
40. [40] — , Functiona of several variables and absolute continuity II, Duke Math. J.6 (1940), 187-215. Zbl0026.39202MR1279JFM66.1224.03
41. [41] — , A eorrection to a previous paper, Duke Math. J.9 (1942), 120-124. Zbl0063.04106MR6560
42. [42] — Multiple integral problems in the calculus of variations and related topica, Univ. of Calif. Publications in Math. new ser.1 (1943), 1-130. Zbl0063.04107MR11537
43. [43] - , The problem of Plateau on a Riemannian manifold, Ann. of Math.49 (1948), 807-851. Zbl0033.39601MR27137
44. [44] — , Quasi-convexity and the lower-semicontinuity of multiple integrate, Pac. J. Math.2 (1952), 25-53. Zbl0046.10803MR54865
45. [45] - , Seoond order elliptic systems of differential equations, Ann. of Math. Studies No. 33Princeton University Press, 1954, 101-159. Zbl0057.08301MR68091
46. [46] — , A variatioval method in the theory of harmonic integrals, IIAmer. J. Math.78 (1956), 137-170. Zbl0070.31402MR87765
47. [47] C.B. Morrey and J. EellsJr., A variational method in the theory of harmonic integrals I, Ann. of Math.63 (1956), 91-128. Zbl0070.09901MR87764
48. [48] M. Nagumo, Uber die gleichmässige Sumierbarkeit und ihre Anwendung auf ein Variationsproblem, Jap. J. Math.6 (1929), 173-182. Zbl55.0156.03JFM55.0156.03
49. [49] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math.80 (1958), 931-954. Zbl0096.06902MR100158
50. [50] O. Nikodym, Sur une classe de fonctions considerées dans l'etude du problème du Dirichlet, Fund. Math.21 (1933), 129-150. Zbl0008.15903JFM59.0290.01
51. [51] G. Nöbeling, Uber die erste Randwertaufgabe bei regularen Variationsproblemen I. Math. Zeit.51 (1949), 712-7.51. Zbl0035.06604MR35400
52. [52] H. Rademacheu, Uber partielle und totale Differenzierbarkeit von Funktionen mehrerer Variabeha, und uber die Transformation der Doppelintegrale, Math. Ann.79 (1918), 340-359. JFM47.0243.01
53. [53] F. Rellich, Ein Satz Uber mittlere Konvergence, Gottingen Nach. (Math.-Phys. Kl.)1930), 30-35, Zbl56.0224.02JFM56.0224.02
54. [54] H. Saks, Theory of the Integral, See. Ed., Warsw-Lwow, 1937; English translation by L. C. Young. Zbl0017.30004JFM63.0183.05
55. [55] — , On the surfaces without tangent planes, Ann. of Math.34 (1933), 114-124. Zbl0006.04903MR1503100JFM59.0289.01
56. [56] L. Schwartz, Theorie des distributions I et II, Actitalites Sci. Ind. 1091 et 1122, Publ. Inst. Mate. Univ. Strasbourg 9 et 10, Paris1950 of 1951. Zbl0042.11405
57. [57] M. Shiffman, Differentiability and analyticity of solutions of double integral variational problemsAnn. of Math.48 (1947), 274-284. Zbl0029.26702MR21256
58. [58] A.G. Sigalov, Conditions for the existence of a minimum of double integrals in un unbounded region, Doklady Akad. Nank SSSR (N.S.)8 (1951), 741-744 (Russian). Zbl0044.10201MR46589
59. [59] — , Regular double integrals of the calculus of variations in non-parametric form, Doklady Akad. Nauk SSSR (N.S.)73 (1950), 891-894 (Russian). MR37478
60. [60] — , Two dimensional problems of the calculus of variations in non-parartretric form, transformed into parametric form, Mat. Sbornik NS94 (76) (1954), 385-406. 
61. [61] — , On conditions of differentiability and analiticity of solutions of two dimensional problems of variations, Doklady Akad. Nauk SSSR (NS)85 (1952), 273-275. MR50185
62. [62] G.I. Silova, Existence of an absolute minimum of multiple integrals in the ealeulus of variations, Doklady Akad. Nank SSSR (N.S.)102 (1955), 699-702. Zbl0064.35301MR70861
63. [63] - , Two dimensional problems of the calculus of variations, Uspehi Matem. Nauk (N.S.)6 (1951), 16-101 (Russian). MR43400
64. [64] S. Sobolev, Sur quelques evaluations concernant les familles des fonctions ayant des derivees a carré integrable, Comptes Rend. Acd, Sci. SSSR. N.S.I.1936, 279-282. Zbl0014.05702JFM62.0266.01
65. [65] - , On a theorem of functional analysis, Mat. Sbornik N.S.4 (1938). 471-497. 
66. [66] D.C. Spfncer, Dirichlet's principle on manifolds, Studies in Mathematics and Mechanics presented to Richard von Mises, Academic Press (1954), 127-134. Zbl0057.07402
67. [67] G. Stampacchia, Sopra una classe di funzioni in due variabili. Applicazioni agli integrali doppi del caleolo delle variazioni, Giorn. Mat. Battaglini (4) 3 (79) (1950), 169-208. Zbl0041.42802MR47932
68. [68] — , Gli integrali doppi del calcolo delle variazioni in forma ordinaria, Atti A-ccad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 8 (1950), 21-26. Zbl0036.07204MR36459
69. [69] — , Sistemi di equazioni di tipo ellittico a derivate parziali del primo ordine e proprietà degli estremi degli integrali multipli, Ricerche Mat.1 (1952), 200-226. Zbl0049.19601MR58141
70. [70] - , Problemi al contorno per equazioni di tipo ellittico e derivate parziali e questioni di calcolo delle variazioni connesse, Ann. Mat. Para Appl. (4) 33 (1952), 211-238 Zbl0047.33902MR51451
71. [71] L. Tonelli, Sulla quadratura delle superficie, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (6) 3 (1926), 633-638. JFM52.0251.02
72. [72] - , Sui massimi e minimi assoluti del calcolo della variazione, Rend. Circ. Mat. Palermo32 (1911), 297-337. Zbl42.0400.01
73. [73] - , Sul caso regolare nel calcolo delle variazioni, Rend. Circ. Mat. Palermo35 (1913), 49-73. Zbl44.0442.03
74. [74] — , Sur une méthode directe du catcul des variations, Rend. Ciro. Mat. Palermo39 (1915), 233-264. Zbl45.0615.02
75. [75] — , La semicontinuità nel calcolo delle variazioni, Rend. Ciro. Mat. Palermo44 (1920), 167-249. 
76. [76] - , Fondamenti del calcolo delle variazioni, Bologna, Zaniohelli, 3 vols. 
77. [77] - , Sur la semi-continuité des intégrales doubles du calcul des variations, Acta Math.53 (1929). 325-346. Zbl55.0899.02JFM55.0899.02
78. [78] - , L'estremo assoluto degli integrali doppi, Ann. Sc. Norm. Pisa (2) 3 (1933), 89-130. Zbl0006.11805JFM59.0500.02
79. [79] L. Van Hove, Sur l'extension de la condition de Legendre du calcul des variations aux integrals multiples a plusieurs fonctions inconnues, Nederl. Akad. Wetensch.50 (1947), 18-23. Zbl0029.26802MR20223

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