Penalization for a PDE with a Nonlinear Neumann boundary condition and measurable coefficients *

Abstract

We consider a system of semi-linear partial differential equations with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized partial differential equations which converges to a solution of our initial problem. The solution we construct is in the L p --viscosity sense, since the coefficients can be not continuous. The method we use is based on backward stochastic differential equations and their S-tightness. The present work is motivated by the fact that many partial differential equations arising in physics have discontinuous coefficients.