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Register a new userThe classical theory of calculus of variations for generalized functions
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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, Jacobis theorem on conjugate points and Noethers theorem. We close with an application to low regularity Riemannian geometry.
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We present an extension of some results of higher order calculus of variations and optimal control 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 the higher order Euler-Lagrange equations, the DAlembert principle in differential form, the du Bois-Reymond optimality condition and the Noethers theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noethers theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais-Uhlenbeck oscillator with singular frequencies.
This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space $mathcal{G}[X,Y]$ of Colombeau generalized functions defined on a manifold $X$ and taking values in a manifold $Y$. This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings $mathcal{C}(X,Y)$ into $mathcal{G}[X,Y]$ and study the sheaf properties of $mathcal{G}[X,Y]$. Similar results are obtained for spaces of generalized vector bundle homomorphisms. Based on these constructions we propose the definition of a space $mathcal{D}[X,Y]$ of distributions on $X$ taking values in $Y$. $mathcal{D}[X,Y]$ is realized as a quotient of a certain subspace of $mathcal{G}[X,Y]$.
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a number of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is how they are still used in informal calculations in Physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, non-linear operations and the generalization of several classical theorems of calculus. Finally, we extend this category of generalized functions into a Grothendieck topos of sheaves over a concrete site. This topos hence provides a suitable framework for the study of spaces and functions with singularities. In this first paper, we present the basic theory; subsequent ones will be devoted to the resulting theory of ODE and PDE.
For each integrability parameter $p in (0,infty]$, the critical smoothness of a periodic generalized function $f$, denoted by $s_f(p)$ is the supremum over the smoothness parameters $s$ for which $f$ belongs to the Besov space $B_{p,p}^s$ (or other similar function spaces). This paper investigates the evolution of the critical smoothness with respect to the integrability parameter $p$. Our main result is a simple characterization of all the possible critical smoothness functions $pmapsto s_f(p)$ when $f$ describes the space of generalized periodic functions. We moreover characterize the compressibility of generalized periodic functions in wavelet bases from the knowledge of their critical smoothness function.
We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frolicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frolicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frolicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Frolicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.
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