Geodesic
Complex calculus of variations
Kybernetika, Tome 39 (2003) no. 2, pp. 249-263
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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.
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.
Classification : 06F05, 30C70, 35F25, 49J10, 49L20, 93B27
Keywords: complex calculus of variation; Hamilton-Jacobi equations 
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Gondran, Michel; Saade, Rita Hoblos. Complex calculus of variations. Kybernetika, Tome 39 (2003) no. 2, pp. 249-263. http://geodesic.mathdoc.fr/item/KYB_2003_39_2_a12/

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