论文标题
具有任意扰动系数扰动的操作员的均质化
Homogenization for operators with arbitrary perturbations in coefficients
论文作者
论文摘要
我们考虑在经典边界条件下的多维域中的一般二阶矩阵操作员。该操作员受到一阶差分运算符的干扰，其系数任意取决于小的多维参数。我们研究了这种扰动操作员的规范分解收敛的意义上的限制（均质化）操作员的存在。我们主要结果的第一部分指出，标准分解收敛等于在乘数的某些空间中扰动操作员中系数的收敛。如果是这种情况，则扰动操作员的分解具有完整的渐近扩展，该扩展均匀地收敛到分解。我们结果的第二部分表明，上述乘数空间中的收敛等同于某些局部平均值在所考虑域的小部分上的收敛性。这些结果由一系列示例支持。我们还提供了一系列生成新的非隔离振动扰动的方法，这最终导致了非常广泛的扰动，我们的结果适用。
We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. This operator is perturbed by a first order differential operator, the coefficients of which depend arbitrarily on a small multi-dimensional parameter. We study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence for such perturbed operator. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multipliers. If this is the case, the resolvent of the perturbed operator possesses a complete asymptotic expansion, which converges uniformly to the resolvent. The second part of our results says that the convergence in the mentioned spaces of multipliers is equivalent to the convergence of certain local mean values over small pieces of the considered domains. These results are supported by series of examples. We also provide a series of ways of generating new non-periodically oscillating perturbations, which finally leads to a very wide class of perturbations, for which our results are applicable.
