Lower Semicontinuous with Lipschitz Coefficients

Abstract

We are interested in integral functionals of the form \begin{equation*} \boldsymbol{J}(U, V) =\int_{\Omega }J\big(x, U(x), V(x)\big) dx, \end{equation*} where $J$ is Carathéodory 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 \begin{equation*} \mathcal{A}(x, \partial)=\sum_{j=1}^{N}A^{(j)}(x)\partial_{x_{j}}, \end{equation*} 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 constant rank. To this end, a framework of paradifferential calculus is needed to deal with the lower smoothness of the coefficients.