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We study generalized group actions on differentiable manifolds in the Colombeau framework , extending previous work on flows of generalized vector fields and symmetry group analysis of generalized solutions. As an application, we analyze group invariant generalized functions in this setting.
We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational invariance are given.
The aim of this paper is to apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of… (More)
We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and… (More)
and Applied Analysis 3 where E is the generalized Young modulus positive constant having dimension of stress , a, b, and c are given positive constants, while 0 < β < α < γ < 1/2. It should be stressed that in 14 there is no discussion concerning restrictions on the parameters a, b, c, α, β, and γ . Instead, only a, b, c ≥ 0, 0 ≤ β < α ≤ 1 and 0 ≤ γ ≤ 1… (More)