Necessary and sufficient conditions for the strong local minimality of C¹ extremals on a class of non-smooth domains

Abstract

Motivated by applications in materials science, a set of quasiconvexity at the boundary conditions is introduced for domains that are locally diffeomorphic to cones. These conditions are shown to be necessary for strong local minimisers in the vectorial Calculus of Variations and a quasiconvexity-based sufficiency theorem is established for C 1 extremals defined on this class of non-smooth domains. The sufficiency result presented here thus extends the seminal theorem by Grabovsky & Mengesha (2009), where smoothness assumptions are made on the boundary.

$\mathcal{A}$-Quasiconvexity, Gårding Inequalities, and Applications in PDE Constrained Problems in Dynamics and Statics

“…Lemma 5 provides some properties of the relative function W (•|•) and its proof can be found in the Appendix. Parts (a)-(c) are collected from [6,7,21].…”

“…It can be seen as a limiting version of a Gårding inequality which replaces the A-quasiconvexity condition in the proof of Theorem 3. Its proof follows [6,7] and relies on an observation in [41] that smooth extremals are spatially local minimisers.…”

“…This adds to the existing results in the case A = curl and bounded domains. Indeed, in Corollary 1, we prove this uniqueness result for A = curl, bounded domains, and free boundary conditions, under the additional assumption of quasiconvexity at the boundary as in [6,7,21]. Section 2, collects all essential definitions and known results in the A-free setting that are used in the following sections.…”

“…the problem of finding (quasiconvexity based) sufficient conditions for a mapȳ to be a strong (or L p ) local minimiser of (1.1), see Section 5. This was indirectly employed in the original proof of [21] and more explicitly in the subsequent proofs in [6,7] which seek the positivity of W (∇ȳ + ∇ϕ|∇ȳ), related to a Gårding-type inequality for the quadratic form W (•|•). In this context, this quadratic form is known as the Weierstrass excess or Efunction, see [7,21] for functionals depending on lower order terms.…”

Corrigendum: Quasiconvex Elastodynamics: Weak‐Strong Uniqueness for Measure‐Valued Solutions

“…In this corrigendum, we reprove Theorem 5.1 ensuring that the constants involved are time-independent. In doing so, we follow a strategy developed by Kristensen and Campos Cordero, see [1], and also Campos Cordero and Koumatos [2]. We also point that in [4] the crucial inequality (1.5) in the proof of Theorem 5.1 below has been obtained with a different proof.…”