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The classical theory of calculus of variations for generalized functions. (English) Zbl 1448.49008

Authors’ abstract: We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

MSC:

49J21 Existence theories for optimal control problems involving relations other than differential equations
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
53B20 Local Riemannian geometry
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