L.Z.(ed.) Gevorkyan’s scientific contributions

... 4] Obviously, the function u(x, y) = u 1 (x, y) + iu 2 (x, y), u k : D −→ R, k = 1, 2, is a solution of Eq. (1.1) if and only if the pair (u 1 , u 2 ) is a solution of the system of partial differential equations { L R u 1 (x, y) = L I u 2 (x, y), L I u 1 (x, y) = −L R u 2 (x, y) ∀(x, y) ∈ D, (1.5) where L R and L I are operators of the type like those in the left hand side of (1.1) with b k := Re b k and b k := Im b k , k = 1, 4, respectively. Equation (1.1) in the improperly elliptic case (cf., e.g., [2, p. 164]) and some Dirichlet-type boundary-value problems for its solutions are considered in [2,Chapter 6] (the boundary ∂D is infinitely differentiable, all boundary functions are infinitely differentiable in ∂D, and solutions are differentiable in the closure of D). The solvability conditions of the Dirichlet boundary-value problem for the improperly elliptic equation (1.1) in the class of the fourth-order continuous differentiable functions in the closure of the unit disk were found by E. Buryachenko (cf., e.g., [3]). ...

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Monogenic functions with values in algebras of the second rank over the complex field and a generalized biharmonic equation with a triple characteristic