Traceless Maps as the Singular Minimizers in the Multi-dimensional Calculus of Variations

Let Ω ⊂ ℝn be a bounded Lipschitz domain and consider the energy functionalover the space of W1,2(Ω, ℝm) where the integrand is a smooth uniformly convex function with bounded second derivatives. In this paper we address the question of regularity for solutions of the corresponding system of Euler–Lagrange equations. In particular, we introduce a class of singularmaps referred to as traceless and examine themas a new counterexample to the regularity of minimizers of the energy functional ℱ[ ·, Ω] using a method based on null Lagrangians.

Constructions of Uniformly Convex Functions

We give precise conditions under which the composition of a norm with a convex function yields a uniformly convex function on a Banach space. Various applications are given to functions of power type. The results are dualized to study uniform smoothness and several examples are provided.