Sobolev regularity for convex functionals on BD

Abstract

We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By RINDLER's foundational work [62], symmetric quasiconvexity is the foremost notion as to sequential weak*-lower semicontinuity of functionals on BD. The overarching main difficulty here is ORNSTEIN's Non-Inequality, hereby implying that the BD-case is genuinely different from the study of variational integrals on BV. In particular, this paper extends the recent work of KRIS-TENSEN and the author [43] from the BV-to the BD-situation. Alongside, we establish partial regularity results for strongly quasiconvex functionals of superlinear growth by reduction to the full gradient case, which might be of independent interest.

The Regularity of Minima for the Dirichlet Problem on BD

“…By our arguments below-and contrary to the W −1,1 -perturbations in the BV-context [14]the correct perturbation space now turns out to be W −2,1 (see Section 2.2.3 for the definition). Without the aforementioned splitting strategy, in turn inspired by Seregin et al [39,74], we would be bound to argue as in [46], and then the desired ellipticity range 1 < a < 1 + 2 n would not be reached. By the degenerate elliptic behaviour of the integrands, non-uniqueness of generalised minima and the overall lack of Korn's inequality, the proof of Theorem 1.1 requires to overcome both technical and conceptual difficulties and is given in Section 4 below.…”

“…Subject to the Dirichlet datum u 0 , the set of all generalised minima is denoted GM(F; u 0 ) and, similarly, the set of all local generalised minima is denoted GM loc (F). As a consequence of [46,Section 5], generalised minimisers always exist in this framework. For future reference, we remark that even if f is strictly convex, generalised minima are not unique in general; see Section 4.1 for more detail.…”

“…In fact, it is known from [14,18] that if 1 < a < 1 + 2 n , then generalised minima of the corresponding full gradient functionals belong to some W 1, p loc with p > 1 whereas in the regime 1 + 2 n a 3, the only W 1,1 -regularity results [14,18] are subject to the additional L ∞ loc -hypothesis on the generalised minima. For variational principles of the form (1.2) subject to (1.4), a first result has been given by Kristensen and the author [46] by passing to fractional estimates. While striving for the optimal ellipticity 1 < a < 1 + 2 n , the method as employed therein only yields the W 1,1 loc -regularity for 1 < a < 1 + 1 n , revealing a crucial ellipticity gap of size 1 n .…”

“…Directly working on the primal problem, we modify and extend the Ekeland viscosity approximation scheme as introduced by Beck and Schmidt [14] in the BV-context and generalised to the BD-situation by Kristensen and the author [46]; see [1,58] for the first applications of the Ekeland variational principle in the regularity context. Here, on the one hand, the Ekeland-type approximations must be strong enough for the (perturbed) Euler-Lagrange equations to permit a splitting strategy, thereby implying the requisite second order estimates for the corresponding almost-minima.…”

“…Next, a lower semicontinuity result in the spirit of [46,Lemma 3.2], [14, Lemma 2.6]: Lemma 2.6. Let 1 < q < ∞, k ∈ N be given and let be open and bounded with Lipschitz boundary ∂ .…”