On domain monotonicity of Neumann eigenvalues of convex domains

Inspired by a recent result of Funano’s, we provide a sharp quantitative comparison result between the first nontrivial eigenvalues of the Neumann Laplacian on bounded convex domains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1 subset-of normal upper Omega 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega _{1} \subset \Omega _{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in any dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> greater than or equal to two, recovering domain monotonicity up to an explicit multiplicative factor. We provide upper and lower bounds for such multiplicative factors for higher-order eigenvalues, and study their behaviour with respect to the dimension and order. We further consider different scenarios where convexity is no longer imposed. In a final section we formulate some related open problems.