Abstract: In response to a challenge delivered by Tonelli in a lecture in Moscow, M. Lavrentiev produced in 1926 an example of a one dimensional variational problem in w hich the infimum of the functional over Lipschitz functions was strictly greater than th e infimum over absolutely continuous functions. This example was later improved by Maniá and others. However it was not until the 1980's that examples were provided involvi ng strictly regular integrands. The existence of such one dimensional examples led Ball and Mizel to raise the question of whether a similar gap phenomenon could arise for mul ti-dimensional boundary value problems involving an elastic material with a physically n atural stored energy function. The present work describes such examples in two dimensio ns. Moreover the stored energy functions are objective and isotropic. The significance of such results for applications is also discussed.
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