  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 ${\rm C}^1$
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.
