SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 1007-1032, January 2021.
We introduce new parametrized classes of shape admissible domains in $\mathbb{R}^n$, $n\geq 2$, and prove that they are compact with respect to the convergence in the sense of characteristic functions, the Hausdorff sense, the sense of compacts, and the weak convergence of their boundary volumes. The domains in these classes are bounded $(\varepsilon,\infty)$-domains with possibly fractal boundaries that can have parts of any nonuniform Hausdorff dimension greater than or equal to $n-1$ and less than $n$. We prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acoustics. A by-product of our proof is the result that the class of bounded $(\varepsilon,\infty)$-domains with fixed $\varepsilon$ is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains.

中文翻译：

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线性声学中的非Lipschitz均匀域形状优化

SIAM控制与优化杂志，第59卷，第2期，第1007-1032页，2021年1月。
我们在$ \ mathbb {R} ^ n $，$ n \ geq 2 $中引入形状可接纳域的新参数化类，并证明它们在特征函数，Hausdorff感知，紧密感，以及边界体积的收敛性较弱。这些类中的域是有界的$（\ varepsilon，\ infty）$域，这些域可能具有分形边界，可以使任何非均匀Hausdorff维数的一部分大于或等于$ n-1 $且小于$ n $。我们证明了在此类中存在最佳形状的情况，可以在线性声学的框架内实现最大的能量耗散。我们的证明的副产品是在Hausdorff会聚下具有固定$ \ varepsilon $的有界$（\ varepsilon，\ infty）$域稳定的结果。