Hölder Estimates for Solutions to a Singular Nonlinear Neumann Problem

Abstract

We consider the elliptic equation $- \Delta u + u = 0$ in a bounded, smooth domain Ω in $\mathbb{R}^n$ subject to the nonlinear singular Neumann condition $\frac{{\partial u}} {{\partial v}} = - u^{ - \beta } + f\left( {x,{\kern 1pt} \;u} \right)$. Here 0 < β < 1 and f ≥ 0 is C¹. We prove estimates for solutions to the same equation with $\frac{{\partial u_\varepsilon }} {{\partial v}} = - \frac{{u_\varepsilon }} {{\left( {u_\varepsilon + \varepsilon } \right)^{1 + \beta } }} + f\left( {x,\;u_\varepsilon } \right)$ on the boundary, uniformly in ε.