Non-Lipschitz Uniform Domain Shape Optimization in Linear Acoustics
| Autor: | Michael Hinz, Anna Rozanova-Pierrat, Alexander Teplyaev |
|---|---|
| Přispěvatelé: | Bielefeld University, Faculty of Mathematics, Bielefeld, Germany, Mathématiques et Informatique pour la Complexité et les Systèmes (MICS), CentraleSupélec, University of Connecticut (UCONN), CentraleSupélec-Université Paris-Saclay |
| Rok vydání: | 2021 |
| Předmět: |
0209 industrial biotechnology
Pure mathematics Control and Optimization mixed boundary value problem FOS: Physical sciences 02 engineering and technology [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] 01 natural sciences Domain (mathematical analysis) Mosco convergence Mathematics - Analysis of PDEs 020901 industrial engineering & automation exten- sions [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Convergence (routing) FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Shape optimization 0101 mathematics Mathematics - Optimization and Control Mathematical Physics Mathematics convergence Applied Mathematics variational 010102 general mathematics variational convergence Mathematical Physics (math-ph) traces Lipschitz continuity uniform domains Functional Analysis (math.FA) Mathematics - Functional Analysis Optimization and Control (math.OC) shape optimization [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] extensions fractal boundaries Analysis of PDEs (math.AP) |
| Zdroj: | SIAM Journal on Control and Optimization SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2021, 59 (2), pp.1007-1032. ⟨10.1137/20M1361687⟩ |
| ISSN: | 1095-7138 0363-0129 |
| Popis: | 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 ($\epsilon$, $\infty$)-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 ($\epsilon$, $\infty$)-domains with fixed $\epsilon$ is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains. |
| Databáze: | OpenAIRE |
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