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# Article

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Keywords:
complex calculus of variation; Hamilton-Jacobi equations
Summary:
In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to ${\mathbf{C}}^n$ functions in ${\mathbf{C}}$. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions to complex Hamilton-Jacobi equations, in particular by generalizing the Hopf-Lax formula.
References:
[1] Balian R., Bloch C.: Solution of the Schrödinger Equation in Terms of Classical Paths. Academic Press, New York 1974 MR 0438937 | Zbl 0281.35029
[2] Evans L. C.: Partial Differential Equations. (Graduate Studies in Mathematics 19.) American Mathematical Society, 1998 MR 1625845
[3] Gondran M.: Convergences de fonctions valeurs dans $\Re ^k$ et analyse Minplus complexe. C.R. Acad. Sci., Paris 1999, t. 329, série I, pp. 783–777 DOI 10.1016/S0764-4442(99)90007-1 | MR 1724540
[4] Gondran M.: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I, pp. 677–680 DOI 10.1016/S0764-4442(01)01901-2 | MR 1842467 | Zbl 1007.49014
[5] Lions P. L.: Generalized Solutions of Hamilton–Jacobi Equations. (Research Notes in Mathematics 69.) Pitman, London 1982 MR 0667669 | Zbl 0497.35001
[6] Voros A.: The return of the quadratic oscillator. The complex WKB method. Ann. Inst. H. Poincaré Phys. Théor. 39 (1983), 3, 211–338 MR 0729194

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