Matthias Bärlin | Searcharxiv

arXiv2022

$\mathcal{A}$-harmonic approximation and partial regularity, revisited

We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,α}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the $\mathrm{L}^{p}$-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.

math.AP↗

arXiv2022

Partial Regularity for $\mathbb{A}$-quasiconvex Functionals

We establish partial Hölder regularity for (local) generalised minimisers of variational problems involving strongly quasi-convex integrands of linear growth, where the full gradient is replaced by a first order homogeneous differential operator $\mathbb{A}$ with constant coefficients. Working under the assumption of $\mathbb{A}$ being $\mathbb{C}$-elliptic, this is achieved by adapting a method recently introduced by Gmeineder (Partial Regularity for Symmetric Quasiconvex Functionals on BD, J. Math. Pures Appl. 145 (2021), Issue 9, pp. 83--129) and Gmeineder & Kristensen (Partial regularity for BV Minimizers, Arch. Ration. Mech. Anal. 232 (2019), Issue 3, pp. 1429--1473).

math.AP↗