On the Dirichlet problem for not strongly elliptic second-order equations

Abstract

1. Let L be a second-order elliptic partial differential operator in C with constant complex coefficients, that is, Lf = af ′′ xx + bf ′′ xy + cf ′′ yy, where a, b, c ∈ C. The ellipticity of L means that the roots λ1 and λ2 of the corresponding characteristic equation aλ + bλ + c = 0 are not real. If L is such that λ1 and λ2 belong to different half-planes of the complex plane with respect to the real line, then L is called strongly elliptic. The classical example of a strongly elliptic operator is the Laplace operator ∆, where ∆f = f ′′ xx + f ′′ yy, while the Bitsadze operator ∂, where ∂f = (f ′ x + if ′ y)/2 is the Cauchy–Riemann operator, serves as an example of a not strongly elliptic one. We denote by C(E) the space of all continuous complex-valued functions on a set E ⊂ C, and put ∥f∥E = supz∈E |f(z)| for f ∈ C(E). A bounded simply connected domain G ⊂ C is said to be regular with respect to the Dirichlet problem for L (or, briefly, L-regular) if for every function h ∈ C(∂G), there is an f ∈ C(G) such that Lf = 0 in G and f ∣∣ ∂G = h. The classical theorem due to Lebesgue [1] states that any bounded simply connected domain G ⊂ C is ∆-regular, that is, it is regular with respect to the classical Dirichlet problem for harmonic functions. For a general strongly elliptic operator L of the form under consideration, there is a conjecture that any bounded simply connected domain is L-regular (see [2], Problem 4.2). This conjecture has been proved only under certain additional fairly restrictive conditions on the regularity of the boundary of G. Namely, the corresponding result was obtained in [3] for Jordan domains with piecewise C-smooth boundary, and this condition on ∂G has not been significantly weakened during the last 20 years. (For instance, the question concerning the L-regularity of an arbitrary Jordan domain G is still open even in the case when ∂G is rectifiable.)