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

中文翻译：

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具有非线性 Neumann 边界条件和可测量系数的 PDE 的惩罚

我们考虑一个具有可测量系数和非线性 Neumann 边界条件的半线性偏微分方程 (PDE) 系统。然后，我们构建了一系列惩罚 PDE，它收敛到我们的初始问题。由于我们考虑的系数可能是不连续的，我们在大号p- 粘度感。我们使用的方法是基于反向随机微分方程及其小号-紧密度。这项工作的动机是物理学中的许多 PDE 具有不连续的系数。因此，如果唯一性成立，则可以通过惩罚来构造解决方案。