平面波从衍射光栅的时谐弹性散射的极限吸收原理
Limiting absorption principle for time-harmonic elastic scattering of plane waves from diffraction gratings
AI总结：
该研究针对周期刚性衍射光栅的时谐弹性散射建立极限吸收原理，通过频率小正虚部扰动正则化不适定问题，得到共振频率下物理解的严格选择机制，框架可推广至多类边界条件。
AI中文摘要：
我们针对周期刚性衍射光栅对平面波的时谐弹性散射，建立了极限吸收原理。通过给频率添加一个小的正虚部进行扰动，我们正则化了传播波数下的不适定问题（即经典瑞利展开条件下唯一性不成立的情况），并利用泛函分析中的奇异扰动结果刻画了极限解。该极限解满足原散射问题，同时附带确保唯一性的附加约束条件。我们同时考虑了入射的压力波和剪切波，两种情形下得到了相同的约束条件。所得结果为共振频率下物理可容许解提供了严格的选择机制。我们的框架可自然推广到诺伊曼（空腔）边界条件及其他传输条件，尤其适用于周期结构中存在导波的情况。
英文摘要：
We establish the limiting absorption principle for time-harmonic elastic scattering of plane waves by a periodic rigid diffraction grating. By perturbing the frequency with a small positive imaginary part, we regularize the ill-posed problem at propagative wavenumbers (that is, when uniqueness fails under the classical Rayleigh expansion condition) and characterize the limiting solution via a singular perturbation result from functional analysis. The limiting solution satisfies the original scattering problem together with an additional constraint that ensures uniqueness. Both incident pressure and shear waves are considered, and the same constraint condition is obtained in both cases. The results provide a rigorous selection mechanism for physically admissible solutions at resonance frequencies. Our framework extends naturally to the Neumann (cavity) boundary condition as well as other transmission conditions, in particular when guided waves exist in periodic structures.
