Interpolating between torsional rigidity and principal frequency
Tom Carroll, Jesse Ratzkin
TL;DR
This paper introduces a family of variational problems that smoothly transition between torsional rigidity and the principal frequency, providing new insights into extremal domains and domain variations.
Contribution
It presents a novel one-parameter interpolation between torsional rigidity and the first Dirichlet eigenvalue, along with derived PDEs and analysis of extremal domains.
Findings
Positive solutions exist in many cases
Extremal domain characterizations are provided
Variations with respect to domain and parameter are analyzed
Abstract
A one-parameter family of variational problems is introduced that interpolates between torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. The associated partial differential equation is derived, which is shown to have positive solutions in many cases. Results are obtained regarding extremal domains and regarding variations of the domain or the parameter.
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