Abstract

We consider a system of second-order semilinear elliptic equations in a multidimensional domain with an arbitrarily curved boundary contained in a narrow layer along the unperturbed boundary. The Dirichlet or Neumann condition is imposed on the curved boundary. In the case of the Neumann condition, rather natural and weak conditions are additionally imposed on the structure of the curving. Under these conditions, we show that the homogenized problem is one for the same system of equations in the unperturbed problem with a boundary condition of the same kind as on the perturbed boundary. The main result is operator \(W_{2}^{1}\) - and L₂- estimates.

中文翻译：

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具有奇异弯曲边界的域中问题的算子估计：狄利克雷和诺依曼条件

摘要

我们考虑多维域中的二阶半线性椭圆方程组，其任意弯曲的边界包含在沿未扰动边界的狭窄层中。狄利克雷或诺依曼条件施加于弯曲边界。在诺伊曼条件的情况下，在弯曲结构上额外施加了相当自然和弱的条件。在这些条件下，我们证明均匀化问题是未扰动问题中同一方程组的问题，其边界条件与扰动边界上的边界条件相同。主要结果是算子\(W_{2}^{1}\) - 和L ₂ - 估计。