2021/09/03 by Kathrin Stollenwerk, Stollenwerk, Kathrin
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2109.01455
openalex publication_date 2021/09/03 · openalex created_date 2021/09/13 · openalex updated_date 2026/07/28
In the 19th century, Lord Rayleigh conjectured that among all clamped plates with given area, the disk minimizes the fundamental tone. In the 1990s, N. S. Nadirashvili proved the conjecture in ℝ2 and M. S. Ashbaugh und R. D. Benguria gave a proof in ℝ2 and ℝ3. In the present paper, we prove existence of an optimal domain for minimizing the fundamental tone among all open and bounded subsets of ℝn, n≥ 4, with given measure. We formulate the minimization of the fundamental tone of a clamped plate as a free boundary value problem with a penalization term for the volume constraint. As the penalization parameter becomes small we show that the optimal shape problem is solved.