精细分层障碍物的散射：频率显式边界和均匀化
Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization
摘要：
我们考虑标量亥姆霍兹方程的变量，不连续系数，建模声波通过各向异性可穿透障碍物的传输。我们首先证明了一个适定性结果和一个频率显式的边界上的解决方案的运营商，与有效的非常大的频率和一类系数，满足一定的单调性条件，在一个空间方向上，只假设是有界的（即，$L^\infty$）。因此，这类系数包括通过具有（可能很大的）数量的层（在2-d中）或纤维（在3-d中）的可穿透障碍物对传输进行建模的系数。重要的是，频率显式的界限保持一致的所有系数在这个类中，这种一致性使我们能够考虑高振荡系数和研究的限制行为时，振荡周期为零。特别是，我们绑定的$H^1$错误所犯的一阶批量校正的均匀化传输问题，与此绑定明确的周期振荡的系数和频率的亥姆霍兹方程;据我们所知，这是第一个均匀化结果的亥姆霍兹方程，这是明确的这两个数量和有效的假设，频率是小的。
英文摘要：
We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the $H^1$ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small.
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