Mathematical Sciences: Partial Regularity in the Calculus ofVariations

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Work is planned on problems in the calculus of variations. Specifically, the principal investigator is concerned with the regularity and existence of minimizers of integral functions whose integrands exhibit a certain type of singular behavior. Fundamental to existence theory is a result of C.B. Morrey which isolates the notion of quasiconvexity as an essential ingredient for existence. A stronger property formulated by L.C. Evans called strict quasiconvexity guarantees partial regularity of minimizers. It has been pointed out in recent literature that in studies of equilibrium boundary value problems in nonlinear elasticity that the growth conditions required to guarantee regularity is too restrictive to include realistic models for hyperelastic materials where the functional being minimized (i.e. its integral) corresponds to stored energy. This problem occurs in particular when one is dealing with compressible materials. Work will be done with integrands exhibiting unboundedness, subject only to the restriction that the determinant of the gradient of admissible functions (representing hydrostatic pressure) be positive. Existence of minimizers is not a problem in this context but regularity is. The variational problem yields an Euler equation but it is not at all clear that a minimizer is even a weak solution of this equation. Work to date suggests nevertheless that a careful analysis of admissible variations will yield the desired regularity. This research is expected to find applications in the study of nonlinear elasticity and incompressible fluid flow.