*Published Paper*

**Inserted:** 15 apr 2023

**Journal:** Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

**Volume:** s. IV, v. 18

**Number:** 3

**Pages:** 321-352

**Year:** 1991

**Links:**
Published paper

**Abstract:**

In this paper, we study two new variational problems involving $S^k$-valued maps, where $S^k = \{ z \in {\mathbb R}^{k+1}, \Vert z \Vert = 1 \} $. We deal with free discontinuity problems since a solution is a pair $(K, u)$, where $K$ is a (a-priori unknown) closed set and $u$ is a map suitably smooth outside of K. These problems can be regarded as a possible schematization of problems in mathematical physics in which both volume forces and surface tensions are present.

**Keywords:**
calculus of variations, free discontinuity problems