Abstract

We give an example of an autonomous functional F(u)=∫Ωf(u,Du)dxF(u) = \int_\Omega f(u, Du)dx (Ω open subset of ℝ² , u : Ω → ℝ² in the Sobolev space W1,1W^{1,1} ) which is sequentially weakly lower semicontinuous in W1,pW^{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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Giovanni Alberti

Istituto di Matematiche Applicate, Universitá di Pisa, via Bonanno 25/B, 56126 Pisa, Italy

Pietro Majer

Dipartimento di Matematica, Universitá di Pisa, via Buonarroti 2, 56127 Pisa, Italy

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Suggested citation

G. Alberti, P. Majer. “Gap Phenomenon for Some Autonomous Functionals.” Journal of Convex Analysis 1 (1994), No. 1, 31–45. https://doi.org/10.68381/jca01002

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Published by Heldermann Verlag, 1994. Rights now held by Banach Press LLC.