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Optimal Shape Design for the p-Laplacian Eigenvalue Problem
Publikationstyp
Journal Article
Date Issued
2019-02-15
Sprache
English
Author(s)
Institut
Journal
Volume
78
Issue
2
Start Page
1231
End Page
1249
Citation
Journal of Scientific Computing 78 (2): 1231-1249 (2019-02-15)
Publisher DOI
Scopus ID
In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach.
Subjects
Eigenvalue optimization
p-Laplacian
Rearrangement algorithm