Article
Ufa Mathematical Journal
Volume 14, Number 4, pp. 14-25
On energy functionals for second order elliptic systems with constant coefficients
Bagapsh A.O., Fedorovskiy K.Yu.
DOI:10.13108/2022-14-4-14
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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.