On the coercivity of elliptic systems in two dimensional spaces

We establish necessary conditions for quadratic forms corresponding to strongly elliptic systems in divergence form to have various coercivity properties in a smooth domain in R². We prove that if the quadratic form has some coercivity property, then certain types of BMO seminorms of the coefficients of the system cannot be very large. We use the connection between Jacobians and Hardy spaces and the special structures of elliptic quadratic forms defined on 2 × 2 matrices.