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Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs. (English) Zbl 1292.35344

Summary: In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by the penalized partial differential equation.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
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