On strongly nonlinar Poincaré boundary value problems for harmonic functions (Q1067094)

Let G be the unit disk in the complex z plane with boundary \(\Gamma =\{t=e^{is}:\) \(s\in [-\pi,\pi]\}\). The following nonlinear Poisson problem is considered: Find a regular harmonic function \(u\in C^ 1(G\cup \Gamma)\) satisfying the boundary condition: \[ \frac{\partial u}{\partial u}+\phi (s,u(e^{is}),\frac{\partial u}{\partial s})=f(s)\quad on\quad \Gamma. \] The following basic assumptions on the data are made: (i) \(\phi\) (s,u,\(\omega)\) is a real-valued continuous function on [- \(\pi\),\(\pi\) ]\(\times {\mathbb{R}}\times {\mathbb{R}}\) which is \(2\pi\)-periodic in s and possesses a continuous partial derivative \(\phi_{\omega}\) and partial derivatives \(\phi_ s\) and \(\phi_ u\) satisfying the Carathéodory conditions and estimations of the form \[ | \phi_ s(s,u,\omega)| \leq E(s)\in L_{\rho}(\Gamma),\quad \rho >1,\quad | \phi_ u(s,u,\omega)| \leq G(s)\in L_{\rho}(\Gamma),\quad \rho >1, \] for u, \(\omega\) from bounded intervals of \({\mathbb{R}}.\) (ii) f(s) is a real-valued absolutely continuous \(2\pi\)-periodic function on [-\(\pi\),\(\pi\) ] possessing a derivative \(f'(s)\in L_{\rho}(\Gamma)\), \(\rho >1.\) The author reduces the problem to an integral equation system to which the classical Schauder fixed point theorem can be applied. Then existence theorems are obtained for some classes of functions \(\phi\) with strong nonlinearities in u and the tangential derivative of u satisfying a constraint on the oscillation of the ascent with respect to the tangential derivative of u and depending in some sense weakly on the function u itself. Some special cases are considered in detail (i.e. \(\phi (s,u,\omega)=cu+\psi (\omega)\); \(\phi (s,u,\omega)=X(u)\psi (\omega)\); \(\phi (s,u,\omega)=\psi (s,u)+\omega X(u):\) the quasilinear case).