African Diaspora Journal of Mathematics

Lower Semicontinuous with Lipschitz Coefficients

Ahmed Zerrouk Mokrane and Mohamed Zerguine

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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.

Article information

Afr. Diaspora J. Math. (N.S.), Volume 10, Number 1 (2010), 55-78.

First available in Project Euclid: 17 May 2010

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Primary: 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]
Secondary: 35D99: None of the above, but in this section 35E99: None of the above, but in this section 76D05: Navier-Stokes equations [See also 35Q30] 35S50: Paradifferential operators 49J45: Methods involving semicontinuity and convergence; relaxation

$\mathcal{A}-$Quasiconvexity Lipschtizian coefficients Pfaffian Young measures paradifferential calculus lower semicontinuous


Mokrane, Ahmed Zerrouk; Zerguine , Mohamed. Lower Semicontinuous with Lipschitz Coefficients. Afr. Diaspora J. Math. (N.S.) 10 (2010), no. 1, 55--78.

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