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By Tarantello G.

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Consider the local super-harmonic (cf. Examples in Section 2) H = δ β− (1 + δ ν ), where ν << 1. 1) in Ωε0 , where C is an arbitrary positive number. , β Lµ (ε0 + ε) − Cε0 − (1 + εν0 ) < min φ. Γε0 Because of the inequality Lµ (ε0 + ε) Lµ (ε0 ) the value C = C(ε0 ) can be chosen independently of ε ∈ [0, 0 ]. With this fixed C we now define the function U = max{uε − CH, φ} in Ωε0 , φ in Dε0 . 4) The function U ε is a global sub-solution for all ∈ [0, 0 ]. Moreover, since H = 0 on ∂Ω and uε is positive in Ωε0 , we have U ε = uε − CH near ∂Ω.

Then there exists c2 > 0 depending only on N , τ , θ, α, c∗ , q0 , C, γ, |Ω|, λn−1 [H], λ, and −1 λn+m [H] such that if δs (φ, φ) c−1 Hw) = m and 2 , then dim ran PG (w PG (H) − PG (w−1 Hw) c2 δs (φ, φ). 2) Proof. We set ρ = 21 dist(λ, (σ(H) ∪ {0}) \ {λ}) and λ∗ = λ if λ is the first nonzero eigenvalue of H, and λ∗ = λn−1 [H] otherwise. 11 (i), it follows that |λk [H] − λk [H]| c(λk [H] + 1)(λk [H] + 1)δ∞ (φ, φ) . This implies that there exists c > 0 such that if δ∞ (φ, φ) < c−1 λk [H]/(λk [H] + 1)2 , then λk [H] 2λk [H].

A. , for example, [15, 18]). ,N is the matrix of coefficients. Here, p0 2 is a constant depending on the regularity assumptions. The best p0 that we obtain is p0 = N which corresponds to the highest degree of regularity (cf. 8), while the case p0 = ∞ corresponds to the lowest degree of regularity in which case only the exponent p = ∞ can be considered. 2. Note that if the coefficients Aij of the operator L are Lipschitz continuous, then δp (φ, φ) does not exceed a constant independent of φ, φ multiplied by the Sobolev norm φ− φ W 1,p (Ω) .

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