Calculus of Variations and Geometric Measure Theory

V. De Cicco - G. Leoni

A chain rule in $L^1(div;A)$ and its applications to lower semicontinuity

created on 14 May 2002
modified by leoni on 10 Jan 2005


Published Paper

Inserted: 14 may 2002
Last Updated: 10 jan 2005

Journal: Calc. Var. Partial Differential Equations
Volume: 19
Number: 1
Pages: 23-51
Year: 2004


A chain rule in the space $L^1(div;A)$ is obtained under weak regularity conditions. This chain rule has important applications in the study of lower semicontinuity problems for functionals of the form $$ intA ( a( x,u) +b( x,u) \cdot D u) +dx,\quad $$ in $W^{1,1}(A)$ with respect to strong convergence in $L^1( A) $ and in turn for general functionals of the form $$ F(u,A):=intA f(x,u(x),D u(x))\,dx, $$ in $W^{1,1}(A)$. Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.

Keywords: Lower Semicontinuity, Chain Rule