| Title:
|
Complex calculus of variations (English) |
| Author:
|
Gondran, Michel |
| Author:
|
Saade, Rita Hoblos |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
39 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
[249]-263 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
complex calculus of variation |
| Keyword:
|
Hamilton-Jacobi equations |
| MSC:
|
06F05 |
| MSC:
|
30C70 |
| MSC:
|
35F25 |
| MSC:
|
49J10 |
| MSC:
|
49L20 |
| MSC:
|
93B27 |
| idZBL:
|
Zbl 1249.49002 |
| idMR:
|
MR1996561 |
| . |
| Date available:
|
2009-09-24T19:53:16Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135525 |
| . |
| Reference:
|
[1] Balian R., Bloch C.: Solution of the Schrödinger Equation in Terms of Classical Paths.Academic Press, New York 1974 Zbl 0281.35029, MR 0438937 |
| Reference:
|
[2] Evans L. C.: Partial Differential Equations.(Graduate Studies in Mathematics 19.) American Mathematical Society, 1998 MR 1625845 |
| Reference:
|
[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 MR 1724540, 10.1016/S0764-4442(99)90007-1 |
| Reference:
|
[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 Zbl 1007.49014, MR 1842467, 10.1016/S0764-4442(01)01901-2 |
| Reference:
|
[5] Lions P. L.: Generalized Solutions of Hamilton–Jacobi Equations.(Research Notes in Mathematics 69.) Pitman, London 1982 Zbl 0497.35001, MR 0667669 |
| Reference:
|
[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 |
| . |
