Whether it is possible to close the gap between the factors of approximation of Box Cover and Orthogonal Polygon Covering has been a long standing open question [KumarR03]. The Minimum Coverage Kernel problem, intermediate between those two, has the…
The following a priori upper bound of λ for positive solutions of (Qλ,ϵ) follows from [13, Proposition 6.1]:
Before proceeding to the proofs of our main results, we prove a partial positivity result on the boundary for nontrivial non-negative solutions of (Pλ):
We now extend our analysis to the upper thresholds. We will show that if λq0 is an eigenvalue of multiplicity m0≥1 of (−Δω), then m0 is a bound for the number of resonances around λq0.
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