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The research of the Dirichlet problem with arbitrary measurable dataforharmonic functions is due to the famous dissertation of Luzin. The presentpaper is devoted to various theorems on the existence of nonclassical solutionsof the Hilbert and Riemann boundary value problems with arbitrary measurabledata for generalized analytic functions by Vekua and the correspondingapplications to the Neumann and Poincare problems for generalized harmonicfunctions. Our approach is based on the geometric (theoretic-functional)interpretation of boundary values in comparison with the classical operatorapproach in PDE.  Here it is proved the existence theorems on solutions of the Hilbert boundaryvalue problem with arbitrary measurable data for generalized analytic functionsin arbitrary Jordan domains with rectifiable boundaries in terms of thena\-tu\-ral parameter and angular (nontangential) limits, moreover, toarbitrary Jordan domains in terms of harmonic measure and principalasymptoticvalues.  Moreover, it is established the existence theorems on solutions for theappropriate boun\-da\-ry value problems of Hilbert and Riemann with arbitrarymeasurable data along the Bagemihl--Seidel systems of Jordan arcs terminatingat the boundary in arbitrary domains whose boundaries consist of finitecollections of rectifiable Jordan curves.  On this basis, it is established the corresponding existence theorems for thePoincare boundary value problem on the directional derivatives and, inparticular, for the Neumann problem with arbitrary measurable data to thePoisson equations.

Neumann and Poincare problems for Poisson's equations with measurable data