Given an arbitrary function we determine the greatest quasiconvex minorant of the function in a way analogous to the classical Legendre-Fenchel transform. The greatest quasiconvex minorant is shown to be the same as the lower semicontinuous regularization of the functional. This fact is used to produce the relaxation of functionals on L ∞ of the form F ( ξ, ξ′ )=ess sup 0≤ s ≤ T h(s, ξ(s), ξ′(s) ). The relaxed functional will be lower semicontinuous in the appropriate topology and yields the existence of a minimizer. Then the relaxation theorem is established, proving that the original problem and the relaxed problem have the same values under broad assumptions on h .
Applied Mathematics and Optimization – Springer Journals
Published: May 1, 1997
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