2018/10/11 by Davide Zucco, Zucco, Davide
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1810.06384
openalex publication_date 2018/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the dependence of optimal constants in Poincar 'e- Sobolev\ninequalities of planar domains on the region where the Dirichlet condition is\nimposed. More precisely, we look for the best Dirichlet regions, among closed\nand connected sets with prescribed total length L (one-dimensional Hausdorff\nmeasure), that make these constants as small as possible. We study their\nlimiting behaviour, showing, in particular, that Dirichler regions homogenize\ninside the domain with comb-shaped structures, periodically distribuited at\ndifferent scales and with different orientations. To keep track of these\ninformation we rely on a \Γ-convergence result in the class of varifolds.\nThis also permits applications to reinforcements of anisotropic elastic\nmembranes. At last, we provide some evidences for a conjecture.\n