  A G\r{a}rding-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 minimisers. Using this new G\r{a}rding
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 minimisers in the classical $\mathcal{A}= {\rm curl}$ case.
