What does this research mean for the field?

The product of the first Dirichlet Laplacian eigenvalue and torsional rigidity can be optimized among sets with prescribed perimeter, and specific values of q > 0 determine whether the ball is a stable local minimizer or maximizer of their weighted product under volume or perimeter constraints. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Investigate relationships between the first Dirichlet Laplacian eigenvalue and torsional rigidity, focusing on their product and power functionals under perimeter and volume constraints.

August 30, 2026Potential AnalysisOpen Access

On Some Functionals Involving Torsional Rigidity, Principal Eigenvalue and Perimeter

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Investigate relationships between the first Dirichlet Laplacian eigenvalue and torsional rigidity, focusing on their product and power functionals under perimeter and volume constraints.
Analyzed shape optimization of the product of the principal eigenvalue and torsional rigidity for sets with prescribed perimeter.
Evaluated functionals across both open sets with finite perimeter and restricted classes of convex domains.
Conducted stability analysis to determine the parameter range where a ball acts as a local extremum for power-weighted product functionals.
Characterized the optimization behavior of the product functional under prescribed perimeter constraints in open and convex sets.
Identified critical exponent thresholds where a spherical ball is a stable local minimizer or maximizer under volume constraints.
Established corresponding local stability and extremum criteria for spherical domains under perimeter constraints.

Implication

Mathematical analysis demonstrates shape optimization conditions for eigenvalue-rigidity functionals in geometric domains, highlighting spherical stability criteria.

Abstract

In this paper we study some relationships between the first Dirichlet Laplacian eigenvalue Λ (Ω ) Λ ( Ω ) and the torsional rigidity T(Ω ) T ( Ω ) of a domain Ω Ω . We consider the problem of optimizing the product Λ (Ω )T(Ω ) Λ ( Ω ) T ( Ω ) among sets with prescribed perimeter, both in the class of open sets with finite perimeter and within the class of convex domains. We also characterize the values of $$q>0$$ q > 0 for which the ball is a stable local minimizer or maximizer of the quantity Λ (Ω )T(Ω )q Λ ( Ω ) T ( Ω ) q , under either a volume or a perimeter constraint.

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Cite This Study

Amato et al. (2026) studied this question.

synapsesocial.com/papers/6a93efc06c1a8fb52e79bd61 https://doi.org/10.1007/s11118-026-10328-2

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Implication

Mathematical analysis demonstrates shape optimization conditions for eigenvalue-rigidity functionals in geometric domains, highlighting spherical stability criteria.

On Some Functionals Involving Torsional Rigidity, Principal Eigenvalue and Perimeter

Key Points

Implication

Abstract

Discussion

Cite This Study

On Some Functionals Involving Torsional Rigidity, Principal Eigenvalue and Perimeter

Key Points

Implication

Abstract

Discussion

Cite This Study

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