Numerical Approximation of the Solution of a Variational Problem with a Double Well Potential

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Abstract
Variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value. Rather, the gradients of minimizing sequences do not converge pointwise and can have oscillations. However, the gradients do converge in the weak topology, i.e., their local spatial averages converge. Such functionals arise in the description of equilibria of crystals or other ordered materials. Stable configurations for solid crystals which have symmetry-related (martensitic) energy wells have a fine-scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit. An analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals.

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