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We consider a boundary value problem in the unit disk for an elliptic equation of order  N  with constant coefficients. We suppose, that the characteristic equation has exactly  M  pairs of complex conjugate roots (0 ≤ 2 M  ≤  N ). The solution of this problem belongs to the class of  N  times continuously differentiable functions, which with up to the ( N  −  M  − 1)-th order derivatives satisfy the Hölder condition in the closed disk. We prove, that the inhomogeneous problem has a solution for arbitrary boundary functions, and the corresponding homogeneous problem (when all boundary functions are identically zero) has ( N  — 2 M ) 2  linearly independent (over the field of real numbers) solutions. The solutions of the homogeneous problem are the purely imaginary polynomials of order 2( N  −  M ) − 2.

On a boundary value problem for an elliptic equation in the unit disk