Representation, relaxation and convexity for variational problems in Wiener spaces

TL;DR

This paper proves convexity of solutions to certain variational problems in Wiener spaces, extending classical Euclidean results through a new representation formula for integral functionals in infinite dimensions.

Contribution

It introduces a novel representation formula for integral functionals in Wiener spaces, enabling the proof of convexity of solutions in this infinite-dimensional setting.

Findings

Convexity of solutions in Wiener spaces established.

Representation formula for integral functionals extended to infinite dimensions.

Results generalize classical Euclidean convexity proofs.

Abstract

We show convexity of solutions to a class of convex variational problems in the Gauss and in the Wiener space. An important tool in the proof is a representation formula for integral functionals in this infinite dimensional setting, that extends analogous results valid in the classical Euclidean framework.