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In the present paper, we consider a boundary value problem with mixed boundary conditions and with a nonlocal parity condition for the Poisson equation in a half-disk. For this problem, we construct Green's function in closed form and use it to write out the solution of the original problem. Unlike [1, p. 194], we need not solve a singular integral equation to obtain the solution. We also consider other boundary conditions and write out the corresponding solutions via Green's function. STATEMENT OF THE PROBLEMwe consider the following problem for the Laplace operator. Find a regular solution of the Poisson equationwith the following boundary conditions. On the semicircle, the parity conditionis imposed and the normal derivativeis given. The directional derivativesare given on the intervals (−1, 0) and (0, 1) of the abscissa axis. For example, the Gellerstedt problem with data on exterior characteristics [2, p. 328] can be reduced to problem (1)-(5). A regular solution of problem (1)-(5) is defined as a function u ∈ C 0 D ∩ C 2 (D), whereD is the closure of D, such that grad u is continuous up to the following parts of the boundary of D : (r = 1, 0 < Θ < π/2) ∪ (y = 0, − 1 < x < 0) ∪ (y = 0, 0 < x < 1). The function f belongs to the Hölder class C α1 D , where 0 < α 1 ≤ 1.For the subsequent exposition, it is convenient to introduce the domain D 1+ = {(r, Θ) : 0 < r < 1, 0 < Θ < π/2}and its closureD 1+ .

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