By Walther H., Voss H.-J.

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G. [26]). In this section we want to apply Λ-theory to construct spaces of generalized functions called ultrafunctions (see also [5, 9–14], and to use them to study a simple class of problems in calculus of variations. As we are going to show, ultrafunctions are constructed by means of a particular version of the hyperfinite approach which can be naturally introduced by means of Λ-theory. In this section we will use the following shorthand notation: for every bounded function ϕ : L → V∞ (R) we let lim ϕ(λ) := lim (λ, ϕ(λ)) .

E. for every φ ∈ C 1 ([0, 1] × Ω) there holds 1 0 Ω 1 ∂t φ dμt dt + Ω 0 ∇φ, vt dμt dt = • we have 1 m 1 0 vt mLm (Ω;μt ) dt Ω φ(1, ·) dρ1 − Ω φ(0, ·) dρ0 ; = Wm (ρ0 , ρ1 ). The other expedient result from Optimal Transport we need is the following convexity property of L q norms. For m = 2, this is a particular instance of a result by McCann, see [17]. 28]. Proposition 2 (Geodesic convexity of L q norms) Let 1 < m < ∞ and let Ω ⊂ R N be an open bounded convex set. Let ρ0 = f 0 · L N and ρ1 = f 1 · L N be two A Note on Some Poincaré Inequalities on Convex Sets by Optimal… 55 probability measures on Ω, such that f 0 , f 1 ∈ L q (Ω) for some 1 ≤ q ≤ ∞.

Nonstandard Analysis: Theory and Applications, NATO ASI series C, vol. 493, pp. 1–49. Kluwer Academic Publishers (1997) 21. : Foundations of Infinitesimal Calculus. University of Wisconsin at Madison (2009) 22. , Wolff, M. ): Nonstandard Analysis for the Working Mathematician. Kluwer Academic Publishers, Dordrecht (2000) 23. : Non-standard Analysis, Lecture Notes, California Institute of Technology. Pasadena (1962) 24. : Internal set theory; a new approach to nonstandard analysis. Bull. Am. Math.

### Uber Kreise in Graphen by Walther H., Voss H.-J.

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