Extremal problems for clamped plates under tension

2025/07/09 by Pedro Freitas, Freitas, Pedro, Roméo Leylekian +1
Computer Science · Engineering · #35G05 #35J35 #35P05 #49R05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Optimization and Control (math.OC) #Topology Optimization in Engineering

paper · pdf · doi:10.48550/arxiv.2507.07040

openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We address extremum problems for spectral quantities associated with operators of the form Δ²-τΔ with Dirichlet boundary conditions, for non-negative values of τ. The focus is on two shape optimisation problems: minimising the first eigenvalue; and maximising the torsional rigidity, both under volume constraint. We establish, on the one hand, a Szegő-type inequality, that is, we show that among all domains having a first eigenfunction of fixed sign the ball minimises the corresponding first eigenvalue; on the other hand a Saint--Venant-type inequality, namely, a sharp upper bound on the torsional rigidity, again achieved by the ball. We further present other properties related to these operators and express the optimality condition associated with the minimisation of the first eigenvalue.

Citations

• Sharp inequalities and asymptotics for polyharmonic eigenvalues
• Fourth order Saint-Venant inequalities: maximizing compliance and mean deflection among clamped plates
• The Bilaplacian with Robin boundary conditions
• Maximum Principle for Higher Order Operators in General Domains
• Spectral analysis of a generalized buckling problem on a ball
• Analyticity and criticality results for the eigenvalues of the biharmonic operator
• The Zero Set of a Real Analytic Function
• A few shape optimization results for a biharmonic Steklov problem
• Optimality conditions for the buckling of a clamped plate
• Nodal properties of eigenfunctions of a generalized buckling problem on\n balls
• Sign-Preserving Property for Some Fourth-Order Elliptic Operators in One Dimension and Radial Symmetry
• Eigenvalues of poly-harmonic operators on variable domains
• Existence of solitary waves in higher dimensions

Related

• On a classical spectral optimization problem in linear elasticity
• Optimality conditions for the buckling of a clamped plate
• Optimization problems involving the first Dirichlet eigenvalue and the\n torsional rigidity
• Upper and lower bounds for eigenvalues of the clamped plate problem
• On the relations between principal eigenvalue and torsional rigidity