Abstract

We present a new extension of a celebrated Serrin's lower semicontinuity theorem. We consider an integral of the calculus of variation ∫Ωf(x,u,Du)dx \int_{\Omega }f\left( x,u,Du\right) dx\, and we prove its lower semicontinuity in Wloc1,1(Ω)W_{loc}^{1,1}\left( \Omega \right) with respect to the strong Lloc1L_{loc}^{1} norm topology, under the usual continuity and convexity property of the integrand f(x,s,ξ)f(x,s,\xi ), only assuming a mild (more precisely, local) condition on the independent variable x∈Rnx\in \Bbb{R}^{n}, say local Lipschitz continuity, which - we show with a specific counterexample - cannot be replaced, in general, by local Hölder continuity.

Author information

Contact details are reproduced from the original publication and may be historical.

Michele Gori

Dip. di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy

gori@mail.dm.unipi.it

Paolo Marcellini

Dip. di Matematica, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy

marcell@math.unifi.it

Keywords

  • Lower semicontinuity
  • strong convergence in L1
  • convex functions
  • local Lipschitz continuity
  • local Hölder continuity
  • calculus of variations

Mathematics Subject Classification

49J45; 35D05

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Suggested citation

M. Gori, P. Marcellini. “An Extension of the Serrin's Lower Semicontinuity Theorem.” Journal of Convex Analysis 9 (2002), No. 2, 475–502. https://doi.org/10.68381/jca09028

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press.