*Published Paper*

**Inserted:** 26 apr 2001

**Last Updated:** 6 may 2003

**Journal:** Journal of Convex Analysis

**Volume:** 2

**Number:** 9

**Pages:** 439-462

**Year:** 2002

**Abstract:**

In this paper we discuss the question of what kind of a
limit can be associated with sequences of variational
problems
$$
\mbox{minimize}\ \ I_{n}(u)\ \ \mbox{s.t.}\ \ \int_{0}^{1
}
u(t)\,dt =x,
$$
associated with integral functionals of the form
$$
I_{n}(u)=\left\{\begin{array}{cl} \displaystyle\int_{0}^{1
}
f_{n}(t,u(t))dt,& \mbox{if the integral makes sense},

\infty,& \mbox{otherwise},\end{array}\right. $$ where on the integrand we basically require only that $f_n(t,u(t))$ be measurable for any measurable $u(t)$.

The main result shows that, even in the
absence of coercivity, there is a functional of the
form
$$
J(\nu)=\int_{0}^{1} \varphi(t,\frac{d\nu}{d\mu}(t))d\mu +
\int_{0}^{1h}(t,\frac{d\nu_{s}{d\nu}_{s}}(t))d

\nu_{s
}
$$
such that for a certain subsequence
\begin{itemize}
\item(a) the liminf inequality of the $\Gamma$-convergence
holds for this functionals and elements of the subsequence;
\item(b) a weaker form of the the limsup inequality also
holds;
\item(c) the value functions of problems (1), (2) for the
functionals of the subsequence $\Gamma$-converge to the
lower closure of the value function of a corresponding
problem for the limit functional.
\end{itemize}

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