On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in unit disk ???? is due to dissertation Luzin. Later on, known monograph Vekua was devoted boundary-value problems only Hölder continuous generalized analytic functions, i.e., complex-valued f(z) complex variable z = x + iy first partial derivatives by Sobolev satisfying equations form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, where complexvalued a; b, and c are assumed belong class Lp some p > 2 smooth enough domains D ?. Our last paper [12] contained theorems on existence nonclassical solutions Hilbert boundaryvalue (with respect logarithmic capacity) f : ? ? such that {\partial}_{\overline{z}}f=g real-valued sources. On this basis, corresponding were established Poincaré directional and, particular, Neumann Poisson ?U G ? Lp; 2, boundary over capacity. present a natural continuation article includes, nonlinear type {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). also Poincar´e ?U(z) H(z)Q(U(z)) investigated us given equations, too. approach based interpretation values sense angular (along nontangential paths) limits conventional tool geometric function theory. As consequences, we give applications concrete semi-linear mathematical physics arising from modelling various physical processes. Those results can be applied anisotropic inhomogeneous media.