SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 561-583, January 2021.
In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with $d$-set boundaries ($N-1\le d<N$). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume).

中文翻译：

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边界对声波的最佳吸收

SIAM控制与优化杂志，第59卷，第1期，第561-583页，2021年1月。
为了找到最简单，最有效的消音墙形状来消散声波的声能，我们考虑了由Helmholtz方程描述的频率模型，该模型在边界处具有阻尼。该模型的适定性在具有$ d $设置边界（$ N-1 \ le d <N $）的一类域中显示。我们引入一类可允许的Lipschitz边界，其中壁的最佳形状在以下意义上存在：我们证明在此形状上存在Radon度量，大于或等于常规Lebesgue度量，为此需要相应的解决方案Helmholtz问题的方程实现了边界上用Lebesgue测度定义的最小声能。如果此Radon量度与Lebesgue量度一致，则相应的解决方案将实现最小的能量。