On energy functionals for second-order elliptic systems with constant coefficients

TL;DR

This paper investigates the existence of energy functionals for certain second-order elliptic systems with constant coefficients, proving that non-reducible strongly elliptic systems do not admit such non-negative quadratic energy functionals.

Contribution

It demonstrates that non-reducible strongly elliptic systems of this type cannot have non-negatively defined quadratic energy functionals, using a reduction to a canonical form.

Findings

Non-reducible strongly elliptic systems lack non-negative energy functionals.

The proof involves reducing systems to a canonical form as perturbations of the Laplace operator.

The result applies to systems with constant coefficients in the Dirichlet problem.

Abstract

We consider the Dirichlet problem for second-order elliptic systems with constant coefficients. We prove that non-reducible strongly elliptic systems of this type do not admits non-negatively defined energy functionals of the form $f\mapsto\int_{D}\varPhi(u_x,v_x,u_y,v_y)\,dxdy$, where $D$ is the domain where the problem we are interested in is considered, $\varPhi$ is some quadratic form in $\mathbb R^4$, and $f=u+iv$ is a function in the complex variable. The proof is based on reducing the system under consideration 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 system under consideration).