DOI: 10.68381/jca01002 ISSN: 0944-6532

Gap Phenomenon for Some Autonomous Functionals

Giovanni Alberti, Pietro Majer
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We give an example of an autonomous functional

F(u) = \int_\Omega f(u, Du)dx F ( u ) = ∫ Ω f ( u , D u ) d x
(Ω open subset of ℝ² , u : Ω → ℝ² in the Sobolev space
W^{1,1} W 1 , 1
) which is sequentially weakly lower semicontinuous in
W^{1,p} W 1 , p
for every p ≥ 1 but does not agree with the relaxation of the same functional restricted to smooth functions when p < 2. A Lavrentiev phenomenon occurs for a related boundary problem.

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