论文标题
具有非线性Neumann边界条件的退化半线性PDE的近似
Approximation of a degenerate semilinear PDEs with a nonlinear Neumann boundary condition
论文作者
论文摘要
我们考虑一个半连接部分微分方程（PDE）的系统，其非线性取决于溶液及其梯度。 Neumann边界条件以非线性方式取决于解决方案。扩散系数不需要均匀的椭圆度。我们表明，这个问题承认可以通过惩罚来近似的粘度解决方案。 Lipschitz条件需要扩散部分的系数。非线性部分以及Neumann条件是Lipschitz。此外，在解变量中假定非线性部分是单调的。请注意，在[13]中已经建立了对该问题的粘度解决方案的存在，然后在[15]中完成。在本文中，我们构建了一个向后向后的随机微分方程（FBSDE）的惩罚系统的序列，然后直接显示其强烈的收敛性。这使我们能够处理非线性取决于解决方案及其梯度的情况。我们的工作尤其扩展了[4]的结果，从某种意义上说，[1，3]的结果。与作品[1、3、4]相反，我们不通过与我们问题相关的随机系统定律的紧凑性薄弱。
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.
