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Variational principles for second-order quasi-linear scalar equations. (English) Zbl 0533.49010

The problem of constructing variational principles for a second-order quasi-linear partial differential equation is considered. In particular, the problem of finding a first-order function f whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian is in the centre of the examination. For such a function f the authors derive two sets of equations. Necessary and sufficient conditions for the integration of the first set are established in the general case, and these lead to a considerable simplification of the second set. In certain special cases, such as the case when the operator is elliptic, the problem is solved completely. The utility of the obtained results is illustrated by a variety of examples.
Reviewer: J.Vaníček

MSC:

49K20 Optimality conditions for problems involving partial differential equations
35A15 Variational methods applied to PDEs
35G20 Nonlinear higher-order PDEs
49L99 Hamilton-Jacobi theories
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[1] Aldersley, S., Higher Euler operators and some of their applications, J. Math. Phys., 20, 522-530 (1979) · Zbl 0416.58028
[2] Anderson, I. M.; Duchamp, T., On the existence of global variational principles, Amer. J. Math., 102, 781-868 (1980) · Zbl 0454.58021
[3] Atherton, R. W.; Homsy, G. W., On the existence and formulation of variational principles for nonlinear differential equations, Studies in Appl. Math., 54, 31-60 (1975) · Zbl 0322.49019
[4] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1967), Wiley-Interscience: Wiley-Interscience New York · Zbl 0365.76001
[5] Courant, R.; Hilbert, D., Methods of Mathematical Physics I (1962), Wiley-Interscience: Wiley-Interscience New York · Zbl 0729.00007
[6] Davis, D. R., The inverse problem in the calculus of variations in a space of \((n + 1)\) dimensions, Bull. Amer. Math. Soc., 35, 371-380 (1929) · JFM 55.0293.05
[7] Debriand, A.; Gaveau, B., Méthode de contrôle optimal en analyse complexe; applications aux spectres et à un problème quasilinéaire elliptique dégénéré, C. R. Acad. Sci. Paris, 285, 1013-1016 (1977) · Zbl 0401.49006
[8] Dedecker, P., On the generalization of sympletic geometry to multiple integrals in the calculus of variations, (Bleuler, K.; Reetz, A., Differential Geometrical Methods in Mathematical Physics. Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics, No. 570 (1973), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg)
[9] Douglas, J., Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc., 50, 71-128 (1941) · JFM 67.1038.01
[10] Finlayson, B. A., The Method of Weighted Residuals and Variational Principles (1972), Academic Press: Academic Press New York · Zbl 0319.49020
[11] Folland, G. B., Introduction to Partial Differential Equations, (Mathematical Notes (1976), Priceton Univ. Press: Priceton Univ. Press Princeton, N.J) · Zbl 0371.35008
[12] Goldschmidt, H.; Sternburg, S., The Hamiltonian-Cartan formalism in the calculus of variations, Ann. Inst. Fourier, 23, 203-269 (1973)
[13] Hartman, P.; Nirenburg, L., On spherical images whose Jacobian does not change size, Amer. J. Math., 81, 901-920 (1959)
[14] Havas, P., The range of application of the Lagrange formalism, I, Nuovo Cimento (Suppl.), 3, 363-388 (1957) · Zbl 0077.37202
[15] Hermann, R., Differential Geometry and the Calculus of Variations, (Interdisciplinary Mathematics, Vol. XVII (1971), Math. Sci. Press: Math. Sci. Press Brookline, Mass) · Zbl 0219.49023
[16] Hermann, R., Geometry, Physics and Systems (1973), Marcel Dekker: Marcel Dekker New York · Zbl 0285.58001
[17] Horndeski, G. W., Differential operators associated with the Euler-Lagrange operator, Tensor, 28, 303-318 (1974) · Zbl 0289.49045
[18] Kuperschmidt, B. A., Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms, (Kaiser, G.; Marsden, J. E., Geometric Methods in Mathematical Physics. Geometric Methods in Mathematical Physics, Lecture Notes in Mathematics, No. 775 (1980), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg) · Zbl 0439.58016
[19] Kwanty, H. G.; Bahar, L. V.; Massimo, F. M., Linear nonconservative systems with asymmetric parameters derivable from a Lagrangian, Hadronic J., 2, 1159-1177 (1979) · Zbl 0432.70031
[20] Mitchell, A. R.; Wait, R., The Finite Element Method in Partial Differential Equations (1977), Wiley-Interscience: Wiley-Interscience New York · Zbl 0344.35001
[21] Nirenberg, L., Lectures on Linear Partial Differential Equations, (Regional Conference Series in Mathematics, Number 17 (1973), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0267.35001
[22] Oden, J. T.; Reddy, J. N., Variational Methods in Theoretical Mechanics (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0324.73001
[23] Olver, P. J., On the Hamiltonian structure of evolution equations, (Proc. Cambridge Philos. Soc., 88 (1980)), 71-90 · Zbl 0445.58012
[24] Olver, P. J.; Shakiban, C., A resolution of the Euler-operator, I, (Proc. Amer. Math. Soc., 69 (1978)), 223-239 · Zbl 0395.49002
[25] Rabinowitz, P. H., A bifurcation theorem for potential operators, J. Funct. Anal., 25, 412-424 (1977) · Zbl 0369.47038
[26] Rabinowitz, P. H., A minimax principle and applications to elliptic partial differential equations, (Chadam, J. M., Nonlinear Partial Differential Equations and Applications. Nonlinear Partial Differential Equations and Applications, Lecture Notes in Mathematics, No. 648 (1978), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg) · Zbl 0152.10003
[27] Rund, H., The Hamilton-Jacobi Theory in the Calculus of Variations (1973), Kreiger: Kreiger New York · Zbl 0264.49016
[28] Shadwick, W. F., The Hamilton-Cartan formalism for \(r\)-th order Lagrangians and the integrability of the KdV and modified KdV equations, Math. Lett. Phys., 5, 137-141 (1981) · Zbl 0473.58012
[29] Shakiban, C., A resolution of the Euler operator II, (Math. Proc. Cambridge Philos. Soc., 89 (1981)), 501-510 · Zbl 0475.49011
[30] Smirnow, M. M., Equations of Mixed Type, (Translations of Mathematical Monographs, Vol. 51 (1978), Amer. Math. Soc: Amer. Math. Soc Providence, R.I)
[31] Takens, F., Symmetries, conservation laws and variational principles, (Geometry and Topology. Geometry and Topology, Lecture Notes in Mathematics, No. 597 (1977), Springer-Verlag: Springer-Verlag Berlin/New York/Heidelberg) · Zbl 0368.49019
[32] Takens, F., A global version of the inverse problem in the calculus of variations, J. Diff. Geometry, 14, 543-563 (1979) · Zbl 0463.58015
[33] Telega, J. J., On variational formulations for non-linear, non-potential operators, J. Inst. Maths. Appl., 24, 175-195 (1979) · Zbl 0417.47029
[34] Tonti, E., Variational formulations of nonlinear differential equations, I, Acad. Roy. Belg. Bull. Cl. Sci., 55, 137-165 (1969) · Zbl 0182.11402
[35] Tonti, E., Variational formulations of nonlinear differential equations, II, Acad. Roy. Belg. Bull. Cl. Sci., 55, 262-278 (1969) · Zbl 0186.14301
[36] Triconi, F. G., Transonic gas flow and equations of mixed type, (Langer, R. E., Partial Differential Equations and Continuum Mechanics (1961), Univ. of Wisconsin Press: Univ. of Wisconsin Press Madison)
[38] Tulczyjew, W. M., The Lagrange differential, Bull. Acad. Polon. Sc. Ser. Sc. Math., Astrong., Phys., 24, 1089-1096 (1976) · Zbl 0352.58002
[39] Tulczyjew, W. M., The Lagrange complex, Bull. Soc. Math. France, 105, 419-431 (1977) · Zbl 0408.58020
[40] Vainberg, M. M., Variational Methods for the Study of Nonlinear Operators (1964), Holden-Day: Holden-Day San Francisco · Zbl 0122.35501
[41] Vinogradov, A. M., A spectral sequence that is connected with a nonlinear differential equation, and the algebraic-geometry foundations of the Lagrange field theory with constraints, Soviet Math. Dokl., 19, 114-118 (1978)
[42] Warner, F. W., Foundations of Differentiable Manifolds and Lie Groups (1971), Scott, Foreman: Scott, Foreman Glenview, Ill · Zbl 0241.58001
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