Calculus of Variations and Geometric Measure Theory

L. Freddi - A. D. Ioffe

On limits of variational problems. The case of a non-coercive functional

created on 26 Apr 2001
modified on 06 May 2003


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


In this paper we discuss the question of what kind of a limit can be associated with sequences of variational problems $$ \mbox{minimize}\ \ In(u)\ \ \mbox{s.t.}\ \ \int01 u(t)\,dt =x, $$ associated with integral functionals of the form $$ In(u)=\left\{\begin{array}{cl} \displaystyle\int01 fn(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 $$ J(\nu)=\int01 \varphi(t,\frac{d\nu}{d\mu}(t))d\mu + \int01h(t,\frac{d\nus}{d
$$ 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}