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Lower Semicontinuous with Lipschitz Coefficients

Volume 10, Number 1 (2010), 55 - 78

Lower Semicontinuous with Lipschitz Coefficients

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Abstract

We are interested in integral functionals of the form
$$
\boldsymbol{J}(U, V) =\int_{\Omega }J\big(x, U(x), V(x)\big) dx,$$
where $J$ is Carath\'eodory positive integrand, satisfying some growth condition of order $p\in(1, \infty)$. We show that $\mathcal{A}(x, \partial)-$quasiconvexity of the integrand $J$ with respect to the third variable is a necessary and sufficient condition of lower semicontinuity of $\boldsymbol{J}$, where $\mathcal{A}(x, \partial)$ is a differential operator given by $$
\mathcal{A}(x, \partial)=\sum_{j=1}^{N}A^{(j)}(x)\partial_{x_{j}},
$$and the coefficients $A^{(j)}, j=1,...,N$ are only Lipschitzian, i.e. $A^{(j)}\in W^{1,\infty }\big(\Omega; \mathbb{M}^{l\times d}\big)$ and satisfy the condition of \textit{constant rank}. To this end, a framework of paradifferential calculus is needed to deal with the lower smoothness of the coefficients.