An Extension of the Serrin's Lower Semicontinuity Theorem

We present a new extension of a celebrated Serrin's lower semicontinuity theorem. We consider an integral of the calculus of variation

and we prove its lower semicontinuity in

W_{loc}^{1,1}\left( \Omega \right) W l o c 1 , 1 ( Ω )

with respect to the strong

L_{loc}^{1} L l o c 1

norm topology, under the usual continuity and convexity property of the integrand

f(x,s,\xi ) f ( x , s , ξ )

, only assuming a mild (more precisely, local) condition on the independent variable

x\in \Bbb{R}^{n} x ∈ R n

, say local Lipschitz continuity, which - we show with a specific counterexample - cannot be replaced, in general, by local Hölder continuity.