# Complex calculus of variations

Michel Gondran; Rita Hoblos Saade

Kybernetika (2003)

- Volume: 39, Issue: 2, page [249]-263
- ISSN: 0023-5954

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topGondran, Michel, and Saade, Rita Hoblos. "Complex calculus of variations." Kybernetika 39.2 (2003): [249]-263. <http://eudml.org/doc/33638>.

@article{Gondran2003,

abstract = {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.},

author = {Gondran, Michel, Saade, Rita Hoblos},

journal = {Kybernetika},

keywords = {complex calculus of variation; Hamilton-Jacobi equations; complex calculus of variation; Hamilton-Jacobi equations},

language = {eng},

number = {2},

pages = {[249]-263},

publisher = {Institute of Information Theory and Automation AS CR},

title = {Complex calculus of variations},

url = {http://eudml.org/doc/33638},

volume = {39},

year = {2003},

}

TY - JOUR

AU - Gondran, Michel

AU - Saade, Rita Hoblos

TI - Complex calculus of variations

JO - Kybernetika

PY - 2003

PB - Institute of Information Theory and Automation AS CR

VL - 39

IS - 2

SP - [249]

EP - 263

AB - 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.

LA - eng

KW - complex calculus of variation; Hamilton-Jacobi equations; complex calculus of variation; Hamilton-Jacobi equations

UR - http://eudml.org/doc/33638

ER -

## References

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- Evans L. C., Partial Differential Equations, (Graduate Studies in Mathematics 19.) American Mathematical Society, 1998 (19.)) MR1625845
- Gondran M., 10.1016/S0764-4442(99)90007-1, C.R. Acad. Sci., Paris 1999, t. 329, série I, pp. 783–777 (1999) MR1724540DOI10.1016/S0764-4442(99)90007-1
- Gondran M., 10.1016/S0764-4442(01)01901-2, C.R. Acad. Sci., Paris 2001, t. 332, série I, pp. 677–680 Zbl1007.49014MR1842467DOI10.1016/S0764-4442(01)01901-2
- Lions P. L., Generalized Solutions of Hamilton–Jacobi Equations, (Research Notes in Mathematics 69.) Pitman, London 1982 Zbl0497.35001MR0667669
- Voros A., The return of the quadratic oscillator, The complex WKB method. Ann. Inst. H. Poincaré Phys. Théor. 39 (1983), 3, 211–338 (1983) MR0729194

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