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A G\r{a}rding-type inequality is proved for a quadratic form associated to$\mathcal{A}$-quasiconvex functions. This quadratic form appears as therelative entropy in the theory of conservation laws and it is related to theWeierstrass excess function in the calculus of variations. The former providesweak-strong uniqueness results, whereas the latter has been used to providesufficiency theorems for local minimisers. Using this new G\r{a}rdinginequality we provide an extension of these results to PDE constrained problemsin dynamics and statics under $\mathcal{A}$-quasiconvexity assumptions. Theapplication in statics improves existing results by proving uniqueness of $L^p$local minimisers in the classical $\mathcal{A}= {\rm curl}$ case.

$\mathcal{A}$-quasiconvexity, G\r{a}rding inequalities and applications in PDE constrained problems in dynamics and statics