... EG0,γrEN0,γr0,completing the proof of the theorem.8 Boundary Value ProblemsRemark 2.3. The restriction a, b ∈W1 in Theorem 2.2 guarantees the existence of such half-eigenvalues, to which the corresponding ... completing the proof of ii.As a function of α,Kα, p is continuous in α ∈ 1, ∞. Explicit formula of Kα, p canbe found in 17, Theorem 4.1. For instance, Kp, pπppand K∞,p2p.Theorem ... BallsWe have proved that for any m ∈ N the in mum HFm,γr can be obtained if only γ ∈ 1, ∞. In the following we will study the property of the minimizers.Theorem 4.5. Given γ ∈ 1, ∞, r>0,...