Approximation of a degenerate semilinear PDEs with a nonlinear Neumann boundary condition

TL;DR

This paper develops a method to approximate solutions of complex semilinear PDEs with nonlinear boundary conditions, using penalized stochastic differential equations, without relying on weak compactness assumptions.

Contribution

It introduces a direct strong convergence approach for penalized FBSDEs to handle nonlinearities depending on both solution and gradient, extending previous results.

Findings

Established existence of viscosity solutions for the PDE system.

Constructed a convergent sequence of penalized FBSDEs.

Extended previous PDE approximation methods to more general nonlinearities.

Abstract

We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required to the diffusion coefficient. We show that this problem admits a viscosity solution which can be approximated by a penalization. The Lipschitz condition is required to the coefficients of the diffusion part. The nonlinear part as well as the Neumann condition are Lipschitz. Moreover, the nonlinear part is assumed monotone in the solution variable. Note that the existence of a viscosity solution to this problem has been established in [13] then completed in [15]. In the present paper, We construct a sequence of penalized system of decoupled forward backward stochastic differential equations (FBSDEs) then we directly show its strong convergence. This allows us to deal with the case where the nonlinearity depends on both the solution and its gradient. Our work extends, in particular, the result of [4] and, in some sense, those of [1, 3]. In contrast to works [1, 3, 4], we do not pass by the weak compactness of the laws of the stochastic system associated to our problem.