Well-Possedness and Iterative Approximation for Elliptic Problems with Nonlinear Logarithmic Robin Boundary Conditions

In this paper, we investigate an elliptic boundary value problem involving a nonlinear logarithmic Robin boundary condition. The problem is formulated on a smooth bounded domain of $\R^2$ with mixed Dirichlet--Neumann--Robin boundary conditions. We introduce an iterative linearization scheme based on a sequence of linear problems and establish the the existence and uniqueness of the weak solution. A rigorous convergence analysis is provided using a local Lipshitz estimate of the logarithmic nonlinearity. We also study, qualitative properties of the solution and validate the theoretical results through numerical experiments based on finite element approximations. Mathematics Subject Classification (2020): 35J25, 35J61, 65N30, 35B45.