On energy functionals for second order elliptic systems with constant coefficients

We consider the Dirichlet problem for second-order elliptic systems with constant coefficients. We prove that non-separable strongly elliptic systems of this type admit no nonnegative definite energy functionals of the form
$$f\mapsto\int\limits_{D}\varPhi(u_x,v_x,u_y,v_y)\,dxdy,$$
where $D$ is the domain in which the problem is considered, $\varPhi$ is some quadratic form in $\mathbb{R}^4$ and $f=u+iv$ is a function of the complex variable. The proof is based on reducing the considered system to a special (canonical) form when the differential operator defining this system is represented as a perturbation of the Laplace operator with respect to two small real parameters, the canonical parameters of the considered system. In particular, the obtained result show that it is not possible to extend the classical Lebesgue theorem on the regularity of an arbitrary bounded simply connected domain in the complex plane with respect to the Dirichlet problem for harmonic functions to strongly elliptic second order equations with constant complex coefficients of a general form is not possible. This clarifies a number of difficulties arising in this problem, which is quite important for the theory of approximations by analytic functions.