The Lavrentiev Gap Phenomenon in Nonlinear Elasticity

The main objective of this paper is to present examples of the Lavrentiev phenomenon within the framework of two-dimensional nonlinear elasticity. Loosely speaking, this phenomenon is associated with the sensitivity of the infimum in a variational problem to the regularity required of the competing mappings. We provide a physically natural stored energy density and reasonable, though nontraditional, boundary conditions such that the energy functional exhibits the Lavrentiev phenomenon with admissible classes that are subsets of the continuous deformations. The stored-energy density W that we produce is smooth, materially homogeneous, frame-indifferent, isotropic and polyconvex. Furthermore, the corresponding minimization problem is such that existence of a continuous minimizer follows from known results.

The basis for our examples is a convex integrand W ₀ for which the Euler-Lagrange equations have a very special form. We show that the functional associated with this W ₀ exhibits the Lavrentiev phenomenon for certain problems; by making a perturbation to W ₀ , we create the stored-energy density W described in the previous paragraph. With other perturbations to the integrand W ₀ and modifications of the boundary conditions, we are able to produce additional examples of the Lavrentiev phenomenon. Finally, we note that the integrand we use is just one of a family of integrands that can be used to produce examples of the phenomenon.