Continuous dependence results for Non-linear Neumann type boundary value problems

TL;DR

This paper establishes estimates on how solutions to certain non-linear Neumann boundary value problems depend continuously on coefficients, extending previous results to more general conditions and domains, and applies these to convergence rates and specific equations.

Contribution

It extends continuous dependence results to more general boundary conditions and domains, including boundary condition dependence, and derives convergence rates for the vanishing viscosity method.

Findings

Derived continuous dependence estimates for non-linear Neumann problems.

Established convergence rates for vanishing viscosity method.

Provided explicit dependence results for Bellman-Isaacs and quasilinear equations.

Abstract

We obtain estimates on the continuous dependence on the coefficient for second order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et.al., Jakobsen-Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman-Isaacs equations and certain quasilinear equation.