The Lavrentiev Phenomenon in Nonlinear Elasticity

In 1985 J.M. Ball and V.J. Mizel raised the question of whether there exist nonlinearly elastic materials possessing a physically natural stored energy density, i.e., one which is independent of an observer's coordinate frame (objective) and is invariant under the group of orthogonal linear transformations of space (isotropic), as well as physically reasonable boundary value problems for such materials such that the infimum of the total stored energy for those continuous deformations of the material meeting the boundary condition (admissible deformations) which belong to a Sobolev space W ^{1 p} 2 for some p ₂>1 is strictly greater than its infimum for those admissible continuous deformations belonging to some Sobolev space W ¹ p ₁, p ₁<p ₂, despite the density of W ^{1 p} 2 in W ¹ p ₁. The question was motivated by M. Lavrentiev's demonstration in 1926 of the presence of such a gap for a 1-dimensional variational boundary value problem on a bounded interval whose smooth integrand satisfied the conditions of Tonelli's existence theorem (as well as the development of improved versions in the 1980's). The present article describes a positive response to the question raised in 1985. Namely, we provide examples of nonlinearly elastic materials in 2-dimensions and physically reasonable boundary value problems for these materials in which a positive gap exists between the infimum of the total stored energy over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 2 and its infimum over admissible continuous deformations belonging to a Sobolev space W ^{1 p} 1, with p ₁<p ₂. The physical and computational significance of such results is also discussed.