Quasi-Convex Integrands and Lower Semicontinuity in $L^1 $
Irene Fonseca, Stefan G. Müller · SIAM Journal on Mathematical Analysis · 1992
In this paper it is shown that, under mild continuity and growth hypotheses, if $f(x,u,.)$ is quasi-convex and if $u_n $, $u \in W^{1,1} $ are such that $u_n \to u$ in $L^1 $, then \[ \int_\Omega {f(x,u(x), abla u(x))dx \leqq \lim \inf } \int_\Omega {f\left( {x,u_n (x), abla u_n (x)} \right)dx.} \] The proof relies on a blowup argument in connection with a truncation result that allows one to consider uniformly convergent sequences.