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Keywords:
convexity; sequential semicontinuity; calculus of variation; minimizer
convexity; sequential semicontinuity; calculus of variation; minimizer
Summary:
The weak lower semicontinuity of the functional $$ F(u)=\int_{\Omega}f(x,u,\nabla u) {\rm d} x $$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on $W^{1,p}(\Omega)$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.
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