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Calculus of Variations: A Differential Form Approach. (arXiv:1712.04896v1 [math.FA])
来源于:arXiv
We study integrals of the form $\int_{\Omega}f\left( d\omega_1 , \ldots ,
d\omega_m \right), $ where $m \geq 1$ is a given integer, $1 \leq k_{i} \leq n$
are integers and $\omega_{i}$ is a $(k_{i}-1)$-form for all $1 \leq i \leq m$
and $ f:\prod_{i=1}^m \Lambda^{k_i}\left( \mathbb{R}^{n}\right)
\rightarrow\mathbb{R}$ is a continuous function. We introduce the appropriate
notions of convexity, namely vectorial ext. one convexity, vectorial ext.
quasiconvexity and vectorial ext. polyconvexity. We prove weak lower
semicontinuity theorems and weak continuity theorems and conclude with
applications to minimization problems. These results generalize the
corresponding results in both classical vectorial calculus of variations and
the calculus of variations for a single differential form. 查看全文>>