Regularity Issues for Cosserat Continua and $p$-Harmonic Maps

Abstract

For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent p from the model is 2 or in ( 32 15 , 3). The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous p-harmonic maps to S 3 . For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model. MSC 2020. 58E20; 74G40; 74B20.

Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality

“…In a recent paper Gastel [6] was exploiting the connection between the theory of Cosserat micropolar elasticity and p-harmonic maps. He showed that the only obstacle for the regularity of minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua is the possible existence of non-constant p-minimizing tangent maps from B 3 into S 3 .…”

Regularity of weak solution of variational problems modeling the Cosserat micropolar elasticity

“…Among many variants and vast body of results of Cosserat theory available in the literature, P. Neff [5,6,7] has made some systematical analysis of the Cosserat theory for micropolar elastic bodies by establishing the existence of minimizers in the framework of calculus of variations. Very recently, in an interesting article [8], Gastel has shown a partial regularity theorem of minimizing weak solutions to a Cosserat energy functional for microplar elastic bodies.…”

“…By extending the techniques in the study of minimizing p-harmonic maps by Schoen-Uhlenbeck [13], Hardt-Lin [9], Fuchs [10], and especially Luckhaus [11], Gastel has recently shown in an interesting article [8] that any minimizer (φ, R) ∈ H 1 (Ω, R 3 )×W 1,p (Ω, SO(3)) of the Cosserat energy functional Coss(φ, R) of the Cosserat functional (1.1) belongs to C 1,α × C α in Ω away from a singular set Σ of isolated points for all 2 ≤ p < 3. Moreover, Σ is shown to be an empty set when p ∈ [2, 32 15 ] by extending stability inequality arguments by Schoen-Uhlenbeck [14], Xin-Yang [15], and Chang-Chen-Wei [16].…”

“…An interesting question to ask is whether the regularity result on minimizers of the Cosserat functional in [8] remains to hold for certain classes of weak solutions to the Cosserat equation (1.2). In this paper, we will answer this question affirmatively.…”

“…) is a nontrivial, stable, stationary p-harmonic map. However, it follows from the stability Lemma 6.3 and Proposition 6.4 in Gastel [8] that for p ∈ (2, 32 15 ), any stable stationary p-harmonic map R…”