*Published Paper*

**Inserted:** 28 sep 2004

**Last Updated:** 2 apr 2007

**Journal:** J. Convex Anal.

**Volume:** 14

**Number:** 1

**Pages:** 69-98

**Year:** 2007

**Abstract:**

In this paper we study the relaxation
with respect to the $L^1$ norm of integral functionals of the type
$$
F(u)=\int_{\Omega} f(x,u,\nabla u)\,dx\quad u\in W^{{1,1}}(\Omega;S^{{d}-1})
$$
where $\Omega$ is a bounded open set of $ R^N$, $S^{d-1}$ denotes the
unite sphere in $ R^d$, $N$ and $d$ being any positive integers,
and $f$ satisfies linear growth conditions in the gradient
variable. In analogy with the unconstrained case, we show that,
if, in addition, $f$ is quasiconvex in the gradient variable and
satisfies some technical continuity hypotheses, then the relaxed
functional $\overline F$ has an integral representation on
$BV(\Omega;S^{d-1})$ of the type
$$
\bar F(u)=\int_{{\Omega}f}(x,u,\nabla
u)\,dx+\int_{{S}(u)}K(x,u^{}-,u^{+,\nu}_{u)\,d{\cal} H}^{{N}-1} + \int_{\Omega} f^{\infty
}
(x,u,d C(u)),
$$
where the suface energy density $K$ is defined by a suitable
Dirichlet-type problem.

**Keywords:**
relaxation, BV functions, unit sphere

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