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In this talk we will discuss minimization problems for
variational integrals of the form $\int_{\Omega} f\left( d\omega \right)$
where $1\leq k\leq n$, $f:\Lambda^{k}(\mathbb{R}^n)\rightarrow\mathbb{R}$
is continuous and $\omega$ is a $\left(k-1\right)$-form. For functions,
this is the classical problem of the calculus of variations where one
studies integrals of the form $\int_{\Omega}f\left( \nabla u \right)$.
We will discuss how the classical framework can be used and extended to
the case of differential forms. We will discuss the appropriate notions
of convexity in this case, generalizing the classical ones and provide
an almost complete picture of their relationships. We shall also see how
this results in an existence theorem to a minimization problem, typically
in the spirit of direct methods in classical calculus of variations.
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