Abstract
We initiate the study of the higher-order Escobar constants I_k(M), kge 3, on bounded planar domains M. The Escobar constants I_k of the unit disk and a family of polygons are provided.
Highlights
Escobar [6] introduced the isoperimetric constant I2(M) of a Riemannian manifold (M, g) with non-empty boundary. In terms of this constant, he gave a lower bound for the first nonzero eigenvalue 1 of the Dirichlet-to-Neumann operator. This theory has been extended by Hassannezhad and Miclo [9] who introduced analogous isoperimetric constants Ik(M) for any k ∈ N+ and used them to provide lower bounds on the higher-order eigenvalues k of the Dirichlet-to-Neumann operator
Escobar constants Let M be a Riemannian manifold with Lipschitz boundary and A(M) denote the family of all non-empty open subsets Ω of M with piecewise smooth boundary Ω, i.e., A(M) = {Ω ⊂ M | Ω ≠, ∂Ω piecewise smooth }
First we show that the Escobar constants Ik(M) are bounded from above by 1
Summary
Escobar [6] introduced the isoperimetric constant I2(M) of a Riemannian manifold (M, g) with non-empty boundary. Main results Escobar [7] and Kusner–Sullivan [13] independently proved that the unit disk maximizes I2 among all bounded domains M in R2 with rectifiable boundary, i.e., I2(M) ≤ I2( ) Conjecture 1 Let M ⊂ R2 be bounded domain with rectifiable boundary for every k≥3 We prove this conjecture for M being a polygon in R2 and k being greater or equal than the number of vertices of M. We generalize these considerations to curvilinear polygons This is done by adapting the proof of Theorem 1.5 in order to find a suitable family of k-tuples near the vertix with smallest angle. We have the inequality Ik(M) ≤ sin( 1∕2), for all k ≥ 3
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