2022/05/16 by Dengler, Marcel
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.07694
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 functional I(u)=∫Ω(1)/(2)|∇ u|2+ρ(det∇ u) dx, where Ω⊂ℝ2 is open and bounded, u∈ W1,2(Ω,ℝ2) and ρ:ℝ→ℝ0+ 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 Wloc2,2. Next, a higher-order regularity result is shown. Indeed, we show that all stationary points that are additionally of class Wloc2,2 and whose Jacobian is suitably Hölder-continuous are of class Cloc∞. As a consequence, these results show that in the case when ‖ρ'‖L^∞(ℝ)<1 all stationary points have to be smooth.