Quadratic variational theory and linear elliptic partial differential equations

Hestenes, Magnus R.

doi:10.1090/S0002-9947-1961-0133575-0

Quadratic variational theory and linear elliptic partial differential equations

Author: Magnus R. Hestenes
Journal: Trans. Amer. Math. Soc. 101 (1961), 306-350
MSC: Primary 35.04; Secondary 35.43
DOI: https://doi.org/10.1090/S0002-9947-1961-0133575-0
MathSciNet review: 0133575
Full-text PDF Free Access

N. Aronszajn, On coercive integro-differential quadratic forms, Conference on Partial Differential Equations, University of Kansas, 1954, Technical Report No. 14, pp. 94-106.

Felix E. Browder, On the regularity properties of solutions of elliptic differential equations, Comm. Pure Appl. Math. 9 (1956), 351–361. MR 90740, DOI https://doi.org/10.1002/cpa.3160090307
Felix E. Browder, Eigenfunction expansions for nonsymmetric partial differential operators. II, Amer. J. Math. 81 (1959), 1–22. MR 107094, DOI https://doi.org/10.2307/2372847
Felix E. Browder, Estimates and existence theorems for elliptic boundary value problems, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 365–372. MR 132913, DOI https://doi.org/10.1073/pnas.45.3.365
J. W. Calkin, Functions of several variables and absolute continuity. I, Duke Math. J. 6 (1940), 170–186. MR 1278

R. Dennemeyer, Quadratic forms in Hilbert space and second order elliptic differential equations, Ph.D. Dissertation, University of California, Los Angeles, 1956, 108 pp.

K. O. Friedrichs, On the differentiability of the solutions of linear elliptic differential equations, Comm. Pure Appl. Math. 6 (1953), 299–326. MR 58828, DOI https://doi.org/10.1002/cpa.3160060301
Lars Gårding, Dirichlet’s problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55–72. MR 64979, DOI https://doi.org/10.7146/math.scand.a-10364
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99

L. Hörmander, On the theory of general partial differential operators, Acta Math. vol. 94 (1955) pp. 161-284.

Magnus R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math. 1 (1951), 525–581. MR 46590
Magnus R. Hestenes, Sufficient conditions for multiple integral problems in the calculus of variations, Amer. J. Math. 70 (1948), 239–276. MR 25091, DOI https://doi.org/10.2307/2372325
Fritz John, Derivatives of continuous weak solutions of linear elliptic equations, Comm. Pure Appl. Math. 6 (1953), 327–335. MR 58829, DOI https://doi.org/10.1002/cpa.3160060302
Peter D. Lax, On Cauchy’s problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615–633. MR 78558, DOI https://doi.org/10.1002/cpa.3160080411
C. B. Morrey Jr., Functions of several variables and absolute continuity, II, Duke Math. J. 6 (1940), 187–215. MR 1279

---, Multiple integral problems in the calculus of variations, University of California Publications in Mathematics, New Series, vol. 1 (1943).

C. B. Morrey Jr., Second-order elliptic systems of differential equations, Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N. J., 1954, pp. 101–159. MR 0068091
Louis Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8 (1955), 649–675. MR 75415, DOI https://doi.org/10.1002/cpa.3160080414
Martin Schechter, On estimating elliptic partial differential operators in the $L_2$ norm, Amer. J. Math. 79 (1957), 431–443. MR 88648, DOI https://doi.org/10.2307/2372690
Martin Schechter, Coerciveness of linear partial differential operators for functions satisfying zero Dirichlet-type boundary data, Comm. Pure Appl. Math. 11 (1958), 153–174. MR 132911, DOI https://doi.org/10.1002/cpa.3160110202
Martin Schechter, Solution of the Dirichlet problem for equations not necessarily strongly elliptic, Bull. Amer. Math. Soc. 64 (1958), 371–372. MR 171071, DOI https://doi.org/10.1090/S0002-9904-1958-10239-6
Martin Schechter, Integral inequalities for partial differential operators and functions satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 37–66. MR 141879, DOI https://doi.org/10.1002/cpa.3160120104
Martin Schechter, General boundary value problems for elliptic partial differential equations, Bull. Amer. Math. Soc. 65 (1959), 70–72. MR 104910, DOI https://doi.org/10.1090/S0002-9904-1959-10279-2

L. van Hove, Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues, Nederl. Akad. Wetensch. Indag. Math. vol. 9 (1947) pp. 3-8.