Penalization for a PDE with a nonlinear Neumann boundary condition and measurable coefficients

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and nonlinear Neumann boundary condition. then construct sequence penalized PDEs, which converges to our initial problem. Since the we may be discontinuous, use notion solution in [Formula: see text]-viscosity sense. The method is based on backward stochastic their S-tightness. This work motivated by fact that many PDEs physics have discontinuous coefficients. As consequence, it follows if uniqueness holds, can constructed penalization.