On coercivity and irregularity for some nonlinear degenerate elliptic systems

We study the ‘universal’ strong coercivity problem for variational integrals of degenerate p-Laplacian type by mixing finitely many homogenous systems. We establish the equivalence between universal p-coercivity and a generalized notion of p-quasiconvex extreme points. We then give sufficient conditions and counterexamples for universal coercivity. In the case of noncoercive systems we give examples showing that the corresponding variational integral may have infinitely many non-trivial minimizers in W01,p which are nowhere C1 on their supports. We also give examples of universally p-coercive variational integrals in W01,p for p ⩾ with L∞ coefficients for which uniqueminimizers under affine boundary conditions are nowhere C1.