# IJPAM: Volume 39, No. 3 (2007)

**FENCHEL TRANSFORMS OF A CONVEX FUNCTIONAL**

Department of Mathematics

University of Florida

P.O. Box 118105, Gainesville, FL 32611, USA

e-mail: yun@math.ufl.edu

e-mail: rao@math.ufl.edu

Department of Mathematics

University of Evansville

1800 Lincoln Avenue, Evansville, IN 47722, USA

e-mail: ct55@evansville.edu

**Abstract.**There are many applications that involve the minimization of a
convex, linear growth function of a measure. For example, image
restoration models, Plateau's problem and deformation of a thin
plate (the plasticity problem) involve minimizing such functions. In
order to understand the theory of these problems, we must understand
how to give meaning , where is a vector-valued measure
and is a convex function with linear growth.

In this paper, we use the space of continuous, bounded functions to
define the Fenchel transform of a function of measure. We then show
that under this definition, the double Fenchel transform coincides
with the definition given by Anzellotti and Giaquinta (1982) and
used throughout the literature. The lower semi-continuity of the
functional is a direct result of properties of the
Fenchel transform.

**Received: **July 4, 2007

**AMS Subject Classification: **49J27, 28C05

**Key Words and Phrases: **function of measure, Fenchel transform

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2007

**Volume:** 39

**Issue:** 3