Aspects of regularity theory in the Calculus of Variations: Integral Estimates and Uniqueness

It has been known for some time that questions about existence and regularity of minimizers for a variational integral on a Dirichlet class are intimately connected to quasiconvexity properties of the corresponding integrand. Quasiconvexity was introduced by Morrey in 1952 in connection with his work on semicontinuity of variational integrals. Under suitable technical assumptions it is essentially equivalent to existence of a minimizer in a given Dirichlet class. In turn, quasiconvexity is defined by requiring minimality of linear mappings over their Dirichlet class. As such it is only marginally more transparent than the problem it was intended to solve. Indeed, it is often very hard to prove or disprove that a given integrand is quasiconvex, and well-known examples of integrands are recorded in the literature where their quasiconvexity (at the origin) would have many interesting ramifications. Strict quasiconvexity, suitably quantified, of the integrand at some point, is essentially equivalent to coercivity of the variational integral on Dirichlet classes. On the other hand, for integrands satisfying the condition for coercivity everywhere (so strongly quasiconvex integrands), Evans derived in 1986 Caccioppoli inequalities of the second kind for the minimizers of the corresponding variational integrals. As a consequence he established ε-regularity results, and hence partial regularity of such minimizers. While the above results are all subject to the condition of quasiconvexity in some form, and therefore can appear superficial, they do gain some traction when it is recalled that quasiconvexity can be framed between rank-one convexity (a necessary condition) and polyconvexity (a sufficient condition).

In these lectures I briefly review the above mentioned results, and illustrate their robustness by casting them in diverse functional set-ups. Particular attention will be paid to coercivity, the connection to integral estimates, and the significance of Caccioppoli inequalities. I then address the questions of full regularity and uniqueness of minimizers under suitable smallness conditions of the Dirichlet boundary datum. It is in this connection interesting to compare the results that can be obtained by use of the Implicit Function Theorem with those that can be obtained using arguments from regularity theory more directly. The last approach yields slightly stronger results, and also raises some interesting questions that involve a Gårding inequality and a Rayleigh type quotient.

The Gårding inequality ensures a certain equi-coercivity of the Rayleigh quotient, and this in turn allows one to prove uniqueness and stability results for minimizers under rather weak smallness conditions on the Dirichlet boundary datum. The validity of the Gårding inequality means that the variational integral is convex on a subspace of finite codimension in the Dirichlet class.

The prerequisites for the lectures are standard undergraduate courses on Measure Theory, Functional Analysis and PDEs.