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Serrin, James

On the definition and properties of certain variational integrals. (English) Zbl 0102.04601

Trans. Am. Math. Soc. 101, 139-167 (1961).
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Keywords:

differentiation and integration, measure theory
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\textit{J. Serrin}, Trans. Am. Math. Soc. 101, 139--167 (1961; Zbl 0102.04601)
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[1] Silvio Cinquini, Condizioni sufficienti per la semicontinuità nel calcolo delle variazioni, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 2 (1933), no. 1, 41 – 58 (Italian). · Zbl 0006.11804
[2] Landolino Giuliano, Sulle condizioni sufficienti per la semicontinuità degli integrali doppi del calcolo delle variazioni, Ann. Scuola Norm. Super. Pisa (2) 10 (1941), 37 – 55 (Italian). · Zbl 0024.32703
[3] Casper Goffman, Lower-semi-continuity and area functionals. I. The non-parametric case, Rend. Circ. Mat. Palermo (2) 2 (1953), 203 – 235 (1954). · Zbl 0052.28601
[4] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. · Zbl 0040.16802
[5] Klaus Krickeberg, Distributionen, Funktionen beschränkter Variation und Lebesguescher Inhalt nichtparametrischer Flächen, Ann. Mat. Pura Appl. (4) 44 (1957), 92, 105 – 133 (German). · Zbl 0082.26702
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[7] Edward James McShane and Truman Arthur Botts, Real analysis, The University Series in Undergraduate Mathematics, D. Van Nostrand Company, Inc., Princeton, N.J.-Toronto-London-New York, 1959. · Zbl 0087.04602
[8] C. B. Morrey, Multiple integral problems in the calculus of variations and related topics, Univ. California Publ. Math., 1943. Especially Chapter III. · Zbl 0063.04107
[9] M. Nagumo, Über die gleichmässige Summierbarkeit und ihre Anwendung auf ein Variationsproblem, Jap. J. Math. vol. 6 (1929) pp. 173-182. See also L. M. Graves, The theory of functions of real variables, New York, McGraw-Hill, 1946, p. 235. · JFM 55.0156.03
[10] C. Y. Pauc, La méthode métrique en calcul des variations, Paris, Hermann, 1941. Especially p. 54. · JFM 67.1036.01
[11] S. Saks, Theory of the integral, 2d ed., Hafner, 1937. · Zbl 0017.30004
[12] Laurent Schwartz, Théorie des distributions. Tome II, Actualités Sci. Ind., no. 1122 = Publ. Inst. Math. Univ. Strasbourg 10, Hermann & Cie., Paris, 1951 (French). · Zbl 0042.11405
[13] James Serrin, On a fundamental theorem of the calculus of variations, Acta Math. 102 (1959), 1 – 22. , https://doi.org/10.1007/BF02559565 James Serrin, A new definition of the integral for nonparametric problems in the calculus of variations, Acta Math. 102 (1959), 23 – 32. · Zbl 0089.08601
[14] James Serrin, On the differentiability of functions of several variables, Arch. Rational Mech. Anal. 7 (1961), 359 – 372. · Zbl 0109.03904
[15] -, On the area of a surface \( z = f(x,y)\), Amer. Math. Monthly vol. 68 (1961) pp. 435-440.
[16] A. G. Sigalov, Two-dimensional problems of the calculus of variations in nonparametric form, Trudy Moskov. Mat. Obšč. 2 (1953), 201 – 233 (Russian). · Zbl 0053.07601
[17] L. Tonelli, Sulla quadratura delle superficie, Rend. Accad. Naz. Lincei (6) vol. 3 (1926) pp. 357-363, 445-450, 633-638. · JFM 52.0251.01
[18] Leonida Tonelli, Sur la semi-continuité des intégrales doubles du calcul des variations, Acta Math. 53 (1929), no. 1, 325 – 346 (French). · JFM 55.0899.02
[19] I. Verchenko, Über das Flächenmass von Mengen, Mat. Sb. (N.S.) vol. 10 (1942) pp. 11-32. · Zbl 0063.08041
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