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

<title>Abstract</title> <p> 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. <bold>Mathematics Subject Classification (2020):</bold> 35J25, 35J61, 65N30, 35B45. </p>