On coercivity and regularity for linear elliptic systems

Abstract

We introduce a nonlinear method to study a ‘universal’ strong coercivity problem for monotone linear elliptic systems by compositions of finitely many constant coefficient tensors satisfying the Legendre–Hadamard strong ellipticity condition. We give conditions and counterexamples for universal coercivity. In the case of non-coercive systems we give examples to show that the corresponding variational integral may have infinitely many nowhere C1 minimizers on their supports. For some universally coercive systems we also present examples with affine boundary values which have nowhere C1 solutions.