2024/06/26 by Raul Fernandes Horta, Horta, Raul Fernandes, Marcos Montenegro +1
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Elasticity and Wave Propagation #FOS: Mathematics #Optimization and Control (math.OC) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2406.18683
openalex publication_date 2024/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The underlying motivation of the present work lies on a cornerstone question in spectral optimization that consists of determining sharp lower and upper uniform estimates for fundamental frequencies of a set of uniformly elliptic operators on a fixed membrane. We solve completely the problem in the plane for the general class of anisotropic operators in divergence form generated by arbitrary norms, which also includes the computation of optimal constants and the characterization of corresponding anisotropic extremizers (if they exist). Our approach is based on an isoanisotropic optimization formulation which, in turn, demands to be addressed within the broader environment of nonnegative, convex and 1-homogeneous anisotropies. A fine and detailed analysis of least energy levels associated to anisotropies with maximum degeneracy leads to a central connection between shapes and fundamental frequencies of rather degenerate elliptic operators. Such a linking also permits to establish that the supremum of anisotropic fundamental frequencies over all fixed-area membranes is infinite for any nonzero anisotropy. As a by-product, the well-known maximization conjecture for fundamental frequencies of the p-Laplace operator is proved for any p other than 2.