The classical theory of calculus of variations for generalized functions

TL;DR

This paper extends classical calculus of variations to generalized functions within a new framework, establishing key theorems and applying it to low-regularity Riemannian geometry.

Contribution

It introduces a generalized smooth functions framework that includes distributions and retains nonlinear properties, extending variational calculus results.

Findings

Established Euler-Lagrange equations for generalized functions

Proved necessary and sufficient conditions for minimizers

Applied theory to low-regularity Riemannian geometry

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.