Problèmes de Neumann quasilinéaires. (Quasilinear Neumann problems) (Q1067093)

The main result is the existence and uniqueness of a solution (u,\(\lambda)\) satisfying \[ (1)\quad -a_{ij}u_{x_ ix_ j}(x)+\phi (x,Du(x))=\lambda \quad in\quad \partial \Omega;\quad \frac{\partial u}{\partial \gamma}=0\quad on\quad \partial \Omega,\quad \int_{\Omega}udx=0. \] Problem (1) is approximated by \[ - a_{ij}u^{\alpha}_{x_ ix_ j}(x)+\phi (x,Du^{\alpha}(x))+\alpha u^{\alpha}(x)=0\quad in\quad \partial \Omega;\quad \frac{\partial u^{\alpha}}{\partial \gamma}=0\quad on\quad \partial \Omega. \] A priori estimates on \(Du^{\alpha}\) are obtained by use of Bernstein-type comparison functions. Then the convergence results are as follows: \[ u^{\alpha}-\frac{1}{| \Omega |}\int_{\Omega}u^{\alpha}dx\to u\quad and\quad \frac{\alpha}{| \Omega |}\int_{\Omega}u^{\alpha}dx\to -\lambda. \]