A concentric ball is a strictly stable local minimizer for the two-phase Robin eigenvalue under volume constraint if and only if the eigenvalue is below the principal Neumann eigenvalue of the inner ball.
Cross-product of Bessel functions: Monotonicity patterns and functional inequalities
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Symmetry-breaking and local stability of a two-phase eigenvalue problem in optimal insulation
A concentric ball is a strictly stable local minimizer for the two-phase Robin eigenvalue under volume constraint if and only if the eigenvalue is below the principal Neumann eigenvalue of the inner ball.