论文标题
线性声学中的非lipschitz统一结构域的形状优化
Non-Lipschitz uniform domain shape optimization in linear acoustics
论文作者
论文摘要
我们在r^n，n $ \ ge $ 2中介绍了新的形状可允许域的形状可允许的域，并证明它们在特征函数，hausdorff的意义上，紧凑的意义，紧凑的意义及其边界量的弱收敛感在融合方面是紧凑的。这些类中的域是有限的（$ε$，$ \ infty $） - 可能是分形边界的域，它们可能具有任何非均匀的Hausdorff尺寸的一部分。我们证明了此类类别中最佳形状的存在，以在线性声学的框架中最大程度地耗散能量。我们证明的副产品是有限级（$ε$，$ \ infty $）的结果 - 固定$ε$的域在Hausdorff Convergence下是稳定的。另一个相关的结果是罗宾型能量功能在收敛域上的MOSCO收敛性。
We introduce new parametrized classes of shape admissible domains in R^n , n $\ge$ 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 ($ε$, $\infty$)-domains with possibly fractal boundaries that can have parts of any non-uniform Hausdorff dimension greater 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 acous-tics. A by-product of our proof is the result that the class of bounded ($ε$, $\infty$)-domains with fixed $ε$ is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains.
