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In this paper we prove an approximation result for the viscosity solution ofa system of semi-linear partial differential equations with continuouscoefficients and nonlinear Neumann boundary condition. The approximation we useis based on a penalization method and our approach is probabilistic. We provethe weak uniqueness of the solution for the reflected stochastic differentialequation and we approximate it (in law) by a sequence of solutions ofstochastic differential equations with penalized terms. Using then a suitablegeneralized backward stochastic differential equation and the uniqueness of thereflected stochastic differential equation, we prove the existence of acontinuous function, given by a probabilistic representation, which is aviscosity solution of the considered partial differential equation. Inaddition, this solution is approximated by solutions of penalized partialdifferential equations.

Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs