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On functions of bounded variation on convex domains in Hilbert spaces

L. Angiuli, S. Ferrari, D. Pallara

Abstract

We study functions of bounded variation (and sets of finite perimeter) on a convex open set $Ω\subseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.

On functions of bounded variation on convex domains in Hilbert spaces

Abstract

We study functions of bounded variation (and sets of finite perimeter) on a convex open set , being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.

Paper Structure

This paper contains 5 sections, 14 theorems, 103 equations.

Table of Contents

  1. Hypotheses and preliminaries
  2. Perturbed Ornstein--Uhlenbeck semigroup on convex domains
  3. BV functions in Hilbert spaces: definitions and some known facts
  4. A De Giorgi type characterisation
  5. Sets of finite perimeter in ${\Omega}$

Key Result

Proposition 1.1

Assume that Hypotheses hyp_base and ipotesi peso are satisfied. Let $p\in(1,\infty)$ and let ${\Omega}$ be an open subset of $X$. The operators $D_H:\mathcal{F}C_b^\infty(\Omega)\rightarrow L^p(\Omega,\nu; H)$ and are closable in $L^p(\Omega,\nu)$ and $L^p(\Omega,\nu)\times L^p(\Omega,\nu)$, respectively. Here $\mathcal{F}C_b^\infty(\Omega)$ is the space of the restrictions to $\Omega$ of functio

Theorems & Definitions (31)

  • Proposition 1.1
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 21 more