On the lower semicontinuity of supremal functionals

In this paper we study the lower semicontinuity problem for a supremal functional of the form F(u,Ω) = ess sup f(x, u(x),Du(x)) with respect to the strong convergence in L ∞ (Ω), furnishing a comparison with the analogoustheory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.

[1]  Gianni Dal Maso,et al.  Integral representation on BV(ω) of Γ-limits of variational integrals , 1979 .

[2]  Giuseppe Buttazzo,et al.  Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations , 1989 .

[3]  G. Aronsson.  Minimization problems for the functional supxF(x, f(x), f′(x)) , 1965 .

[4]  Gunnar,et al.  Minimization problems for the functional supxF(x, f(x), f'(x)). (II) , 1966 .

[5]  Carlo Sbordone,et al.  Some properties of Γ-limits of integral functionals , 1979 .

[6]  Guy Bouchitté,et al.  Homogenization of elastic thin structures: a measure fattening approach , 2002 .

[7]  B. Brunt.  The calculus of variations , 2003 .

[8]  G. Buttazzo,et al.  On the lower semicontinuity of certain integral functionals , 1983 .

[9]  G. Aronsson.  Extension of functions satisfying lipschitz conditions , 1967 .

[10]  L. Berkovitz.  Lower semicontinuity of integral functionals , 1974 .

[11]  G. Buttazzo,et al.  The Class of Functionals which can be Represented by a Supremum , 2002 .

[12]  Goswin Eisen.  A counterexample for some lower semicontinuity results , 1978 .

[13]  I. Fonseca,et al.  On lower semicontinuity and relaxation , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[14]  E. Barron,et al.  Calculus of variations inL∞ , 1997 .

[15]  E. N. Barron,et al.  Lower semicontinuity of L∞ functionals , 2001 .

[16]  Paolo Marcellini,et al.  An Extension of the Serrin's Lower Semicontinuity Theorem , 2002 .

[17]  I. Fonseca,et al.  Some remarks on Lower semicontinuity , 2000 .

[18]  James Serrin,et al.  On the definition and properties of certain variational integrals , 1961 .

[19]  E. N. Barron,et al.  Calculus of variations in L , 1997 .

[20]  A. Ioffe.  On Lower Semicontinuity of Integral Functionals. II , 1977 .