Everywhere regularity results for a polyconvex functional in finite elasticity

Abstract

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

On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory

“…By the higher-order regularity result [17,Thm 1.4] we know that u = re M R ∈ W 2,2 (B, R 2 ) and d ∈ C 1 ([0, 1]), is enough to imply full local smoothness u = re M R ∈ C ∞ loc (B, R 2 ) which guarantees smoothness at the origin, the smoothness can then be extended by limit taking of r, d and their derivatives up to the harmless boundary at R = 1, showing the claim. Now for the second part of the statement we first note that ρ(d) > 0 for all d > 0 implies that there exists a δ > 0 s.t.…”

“…), but not necessarily any better, and r(0 What does the literature tell us about the regularity of stationary points/minimizers of the energy I? This functional has recently been introduced in [17]. There it is shown that for arbitrary γ ∈ (0, ∞) any stationary point of the energy I subject to arbitrary L 2 − boundary data needs to be locally Hölder-continuous and a higher-order regularity result is obtained.…”