Integral Functionals on Lp-Spaces: Infima Over Sublevel Sets

ON APPROXIMATION OF LINEAR FUNCTIONALS ON Lp SPACES

In a recent paper certain approximations to continuous nonlinear functionals de-ned on an L p space (1 < p < 1) are shown to exist. These approximations may be realized by sigmoidal neural networks employing a linear input layer that implements nite sums of integrals of a certain type. In another recent paper similar approximation results are obtained using elements of a general class of contin...

Representation of nonlinear transformations and functionals on LP spaces

Estimates on Path Functionals over Wasserstein Spaces

In this paper we consider the class a functionals (introduced in [BBS]) Gr,p(γ) defined on Lipschitz curves γ valued in the p-Wasserstein space. The problem considered is the following: given a measure μ, give conditions in order to assure the existence a curve γ such that γ(0) = μ, γ(1) = δx0 , and Gr,p(γ) < +∞. To this end, new estimates on Gr,p(μ) are given and a notion of dimension of a mea...

Perimeter of sublevel sets in infinite dimensional spaces

We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter–example, we show that in general this is not true for compact convex domains.

Integral functionals on Sobolev spaces having multiple local minima

THEOREM A. Let (X, τ) be a Hausdorff topological space and Ψ : X →]−∞,+∞], Φ : X → R two functions. Assume that there is r > infX Ψ such that the set Ψ (]−∞, r]) is compact and first-countable. Moreover, suppose that the function Φ is bounded below in Ψ(]−∞, r]) and that the function Ψ+ λΦ is sequentially lower semicontinuous for each λ ≥ 0 small enough. Finally, assume that the set of all glob...