# $\mathcal{A}$-quasiconvexity, Gårding inequalities, and applications in PDE constrained problems in dynamics and statics

Koumatos, Konstantinos and Vikelis, Andreas P (2021) $\mathcal{A}$-quasiconvexity, Gårding inequalities, and applications in PDE constrained problems in dynamics and statics. SIAM Journal on Mathematical Analysis, 53 (4). pp. 4178-4211. ISSN 0036-1410

A Gårding-type inequality is proved for a quadratic form associated to $\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimizers. Using this new Gårding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$-local minimizers in the classical $\mathcal{A}={curl}$ case.