Here we develop a regularity theory for a polyconvex functional in 2 × 2−dimensional compressible finite elasticity. In particular, we consider energy minimizers/stationary points of the functionalwhere Ω ⊂ R 2 is open and bounded, u ∈ W 1,2 (Ω, R 2 ) and ρ : R → R + 0 smooth and convex with ρ(s) = 0 for all s ≤ 0 and ρ becomes affine when s exceeds some value s0 > 0. Additionally, we may impose boundary conditions.The first result we show is that every stationary point needs to be locally Hölder-continuous. Secondly, we prove that if ρ ′ L ∞ (R) < 1 s.t. the integrand is still uniformly convex, then all stationary points have to be in W 2,2 loc . Next, a higher-order regularity result is shown. Indeed, we show that all stationary points that are additionally of class W 2,2 loc and whose Jacobian is suitably Hölder-continuous are of class C ∞ loc . As a consequence, these results show that in the case when ρ ′ L ∞ (R) < 1 all stationary points have to be smooth.2020 Mathematics Subject Classification. 49N60, 73C50. Key words and phrases. Calculus of Variations, elasticity, polyconvexity, regularity.1 It would be enough to assume ρ ∈ C k for some k ≥ 2 with the necessary changes in the statements that (M. Dengler

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In this paper we consider the problem of minimizing functionals of the form E(u) = B f (x, ∇u) dx in a suitably prepared class of incompressible, planar maps u : B → R 2 . Here, B is the unit disk and f (x, ξ) is quadratic and convex in ξ. It is shown that if u is a stationary point of E in a sense that is made clear in the paper, then u is a unique global minimizer of E(u) provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional f (x, ξ), depending smoothly on ξ but discontinuously on x, whose unique global minimizer is the so-called N −covering map, which is Lipschitz but not C 1 .√ N e R (N θ), where N ∈ N \ {0} and 0 ≤ θ ≤ 2π.