Bibliography
P. Alexandroff and H. Hopf,Topologie I, Julius Springer, Berlin, 1935.
J. W. Calkin,Functions of several variables and absolute continuity, I, Duke Math. J. 6 (1940), 170–186.
L. Cesari,Una uguaglianza fondamentale per l'area delle superficie, Atti Reale Accademia Italia 14 (1944), 891–950.
L. Cesari,La nozione di integrale sopra una superficie in forma parametrica, Ann. Scuola Norm. Sup. Pisa 13 (1947), 77–117.
L. Cesari,Condizioni sufficienti per la semicontinuità degli integrali sopra una superficie in forma parametrica, Ann. Scuola Norm. Sup. Pisa 14 (1948), 47–79.
L. Cesari,An existence theorem for integrals on parametric surfaces, Amer. J. Math. 74 (1952), 265–295.
L. Cesari,Surface Area, Princeton University Press, Princeton, 1956.
R. Courant,Dirichlet's Principle, Interscience Publishers, New York, 1950.
J. M. Danskin,On the existence of minimizing surfaces in parametric double integral problems in the calculus of variations, Riv. Mat. Univ. Parma 3 (1952), 43–63.
H. Lewy,On minimal surfaces with partially free boundary, Comm. Pure Appl. Math. 4 (1951), 1–13.
C. B. Morrey, Jr.,An analytic characterization of surfaces of finite lebesgue area, I, Amer. J. Math. 57 (1935), 692–702.
C. B. Morrey, Jr.,Functions of several variables and absolute continuity, II, Duke Math. J. 6 (1940), 187–215.
C. B. Morrey, Jr.,Multiple integral problems in the calculus of variations and related topics, Univ. California Publ. Math., N. S. Vol. 1, 1943, 1–130.
C. B. Morrey, Jr.,The parametric variational problem for double integrals, Comm. Pure Appl. Math. 14 (1961), 569–575.
C. B. Morrey, Jr.,Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966.
C. B. Morrey, Jr. and J. Eells, Jr.,A variational method in the theory of harmonic integrals, I, Ann. of Math. 63 (1956), 91–128.
A. G. Sigalov,Two dimensional problems in the calculus of variations, Uspehi Matem. Nauk. N. S., 6 (1951), 16–101. English: Translation 83, Amer. Math. Soc., 1953.
A. G. Sigalov,Variational problems with admissible surfaces of arbitrary topological types, (Russian) Uspehi Matem. Nauk. 12, (1957), 53–98.
S. L. Sobolev,Applications of Functional Analysis in Mathematical Physics, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, 1963.
L. Tonelli,Sul problema di Plateau, I and II, Rend. Accad. Naz. Lincei, (6) 29 (1936), 335–339, 393–398.
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This research was supported in part by N.S.F. Grants GP-57 and GP-3920 while the author was at the University of Michigan. This is a new version of a portion of the author's Doctoral Dissertation written at the University of Michigan under the direction of Professor Lamberto Cesari. Author now at the University of Kansas, Lawrence, Kansas (U.S.A.).
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Lipkin, L.J. A free boundary problem for parametric integrals of the calculus of variations. Rend. Circ. Mat. Palermo 17, 33–67 (1968). https://doi.org/10.1007/BF02849549
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DOI: https://doi.org/10.1007/BF02849549