Higher gradient integrability of minimizers for functionals with polyconvex local energies

Michael Matthias Dougherty, Purdue University

Abstract

In this work I show that minimizers and other equilibrium points of certain classes of functionals in the calculus of variations are in higher Sobolev spaces than existence theory provides. In particular I show that the gradients of such equilibria are locally integrable to any positive finite power. It follows that these functions themselves are in fact Holder continuous. The functionals considered here have local energies which are polyconvex, or more generally quasiconvex, functions of the gradient. I present in Chapter 1 the relevant definitions and a survey of previous results. The highlight of Chapter 1 is a result of DiBenedetto and Manfredi, which gives gradient estimates for solutions to nonhomogeneous p-Laplace equations. This result is utilized in the proofs of the new results in Chapter 2. In Chapter 2, Section 2.2 I present a joint result with Daniel Phillips in which we show that equilibrium points, i.e., functions which satisfy the Euler-Lagrange equations in the sense of distributions, have this higher integrability of the gradient when the local energy is a quasiconvex perturbation of a power of the gradient. There we assume polynomial growth of the perturbation strictly less than that of the gradient term. In Section 2.3 I prove the same conclusion when the local energy is the sum of norm squared of the gradient and a convex, asymptotically linear function of the Jacobian. Thus the local energy is the sum of a convex and a polyconvex function of the gradient. The result holds even if the polyconvex term dominates the convex term in certain directions. Throughout Chapter 2, I set these new results in context with the previous results and the theory of nonlinear elasticity.

Degree

Ph.D.

Advisors

Phillips, Purdue University.

Subject Area

Mathematics

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