Dirichlet problem for elliptic equations and systems with constant coefficients on the plane

The Dirichlet problem for second-order elliptic systems with constant coefficients in ${\mathbb R}^2$ is considered. For inseparable strongly elliptic systems of the indicated type, the problem of the existence of non-negative definite energy functionals of the form
$$ f\mapsto\int_{D}\varPhi(u_x,v_x,u_y,v_y)\,dxdy, $$
where $D$ is the domain in which the problem is considered, $\varPhi$ is a quadratic form in $\mathbb R^4$, and $f=u+iv$ is a function of a complex variable.