*Published Paper*

**Inserted:** 26 dec 2001

**Journal:** Journal of Convex Analysis

**Volume:** 7

**Number:** 2

**Pages:** 271-297

**Year:** 2000

**Abstract:**

Starting from the results of DE ARCANGELIS R., TROMBETTI C.: {\it
On the Relaxation of Some
Classes of Dirichlet Minimum Problems}; Comm. Partial Differential
Equations *24*, (1999), 975-1006, the Lavrentieff
phenomenon for the functional
$F(\Omega,\varphi_O,\cdot)\colon u\in BV(\Omega)\mapsto
\int_\Omega f(\nabla u)dx + \int_\Omega f^\infty({dD^su\over
d

D^su

})d

D^su

+\int_{\partial\Omega}
f^\infty((\varphi_0-\gamma_\Omega (u))*n*) d{\cal H}^{n-1}$
between
$BV(\Omega)$ and
$BV(\Omega)\cap C^1(\Omega)$ is studied, where $f\colon *R*^n\to [0,+\infty[$ is convex, $f^\infty$ is its recession
function,
$\varphi_0\in L^1(\partial\Omega)$, and
$\gamma_\Omega$ is the trace operator on $\partial\Omega$. The
occurrence of the phenomenon is first discussed by means
of an example, and then completely characterized. Sufficient
conditions implying the absence of the phenomenon are also
proved, and some relaxation properties of
$F(\Omega,\varphi_O,\cdot)$ are also established.