  We present a class of models of elastic phase transitions with incompatible
energy wells in any space dimension, where an abundance of Lipschitz global
minimizers in a hard device coexists with a complete lack of strong local
minimizers. The analysis hinges on the proof that every strong local minimizer
in a hard device is also a global minimizer which is applicable much beyond the
chosen class of models. Along the way we show that a new proof of sufficiency
for a subclass of affine boundary conditions can be built around a novel
nonlinear generalization of the classical Clapeyron theorem, whose subtle
relation to dynamics was studied extensively by R. Fosdick.
