Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup

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SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “It is well known that the spectrum of the Steklov problem is discrete as long as the trace operator H(1)(Ω)→ L(2)(∂Ω) is compact (see [7]).”
Local Lp-Brunn-Minkowski inequalities for p<1 (2017, methodology): “ the even function u(x) = |x| 2 2 , note that: Z ∂K u2 ν hx,ν∂K(x)i dx = Z ∂K hx,ν∂K(x)idx = Z K div(x)dx = nV(K) = Z K ∇2u 2 dx. It follows by deﬁnition that D(K) ≥ B(K) ≥ 1. Lemma 6.12. For any q ∈ [1,2], 1 ≤ D(Bn q ) ≤ C. For any q ∈ [2,∞], D(Bn q ) ≤ Cn1− 2 q . Proof. The lower estimate is given”
Eventually positive semigroups of linear operators (2015, background): “A proper construction of Dλ as the generator of a strongly continuous analytic semigroup on L(2)(∂Ω) can be found in [4, 7].”
Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, background): “We do not know how to handle the range q ∈ [1, 2) in Theorem 1.”