Optimization of the First Dirichlet Laplacian Eigenvalue with Respect to a Union of Balls

• Published in: The Journal of geometric analysis 2023-06, Vol.33 (6), Article 184
• Main Authors: Birgin, E. G., Fernandez, L., Haeser, G., Laurain, A.
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Algorithms; Convex and Discrete Geometry; Differential Geometric PDE Control; Differential Geometry; Dirichlet problem; Dynamical Systems and Ergodic Theory; Eigenvalues; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Numerical analysis; Optimization; Polygons; Sensitivity analysis; Shape Optimization and Applications
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01241-w

The problem of minimizing the first eigenvalue of the Dirichlet Laplacian with respect to a union of m balls with fixed identical radii and variable centers in the plane is investigated in the present work. The existence of a minimizer is shown and the shape sensitivity analysis of the eigenvalue with respect to the centers’ positions is presented. With this tool, the derivative of the eigenvalue is computed and used in a numerical algorithm to determine candidates for minimizers. Candidates are also constructed by hand based on regular polygons. Numerical solutions contribute in at least three aspects. They corroborate the idea that some of the candidates based on regular polygons might be optimal. They also suggest alternative regular patterns that improve solutions associated with regular polygons. Lastly and most importantly, they delivered better quality solutions that do not follow any apparent pattern. Overall, for low values of m , candidates for minimizers of the eigenvalue are proposed and their geometrical properties as well as the appearance of regular patterns formed by the centers are discussed.