Extremal Rearrangement Problems Involving Poisson’s Equation with Robin Boundary Conditions

Abstract

In this paper, we study both minimization and maximization problems corresponding to a Poisson’s equation with Robin boundary conditions. These rearrangement shape optimization problems arise in many applications including the design of mechanical vibration and fluid mechanics that explore the possibility to control the total displacement and the kinetic energy, respectively. Analytically, we study the properties of the extremizers on general domains including topology and geometry of the optimizers. Asymptotic behaviors of the optimal values are investigated as well. Although the explicit solutions are rare for this kind of optimization problems, we obtain such solutions on N-balls. Numerically, we propose efficient algorithms based on finite element methods, rearrangement techniques and our analytical results to determine the extremizers in just a few iterations on general domains.

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Acknowledgements

Authors would like to thank Mathematics division, National Center of Theoretical Sciences, Taipei, Taiwan for hosting a research pair program during June 15-June 30, 2019 to support this study.

Funding

Chiu-Yen Kao is partially supported by NSF grant DMS-1818948.

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This work is partially supported by NSF Grant DMS-1818948.

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Kao, CY., Mohammadi, S.A. Extremal Rearrangement Problems Involving Poisson’s Equation with Robin Boundary Conditions. J Sci Comput 86, 40 (2021). https://doi.org/10.1007/s10915-021-01413-2

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Keywords

  • Poisson’s equation
  • Shape optimization
  • Rearrangement
  • Mechanical vibration
  • Robin boundary conditions

Mathematics Subject Classification

  • 35Q93
  • 49J20
  • 49M99
  • 35J20
  • 74E30