Weighted Hardy inequality with higher dimensional ... - CAPDE

capde.cl

Weighted Hardy inequality with higher dimensional ... - CAPDE

where c is a positive constant depending only on Ω, p, q, η and Σ k . Hence by (3.7)we concludeµ λ (Ω, Σ k ) ≤ 1 + r1 − cr1 + cε1 − cε( (N − k)24)+ τ + cr1 − r .Taking the limit in ε, then in r and then in τ, the claim follows.Step 2: We claim that there exists ˜λ ∈ R such that µ˜λ(Ω, Σ k ) ≥ (N−k)24.Thanks to Lemma 3.1, the proof uses a standard argument of cut-off function andintegration by parts (see [2]) and we can obtain∫∫∫δ −2 u 2 q dx ≤ |∇u| 2 p dx + C δ −2 u 2 η dx ∀u ∈ Cc ∞ (Ω),ΩΩfor some constant C > 0. We skip the details. The claim now follows by choosing˜λ = −CΩFinally, noticing that µ λ (Ω, Σ k ) is decreasing in λ, we can set}(3.9) λ ∗ (N − k)2:= sup{λ ∈ R : µ λ (Ω, Σ k ) =4so that µ λ (Ω, Σ k ) < (N−k)24for all λ > λ ∗ .4 Non-existence resultLemma 4.1 Let Ω be a smooth bounded domain of R N , N ≥ 3, and let Σ k be asmooth closed submanifold of ∂Ω of dimension k with 1 ≤ k ≤ N − 2. Then, thereexist bounded smooth domains Ω ± such that Ω + ⊂ Ω ⊂ Ω − and∂Ω + ∩ Ω = ∂Ω − ∩ Ω = Σ k .Proof. Consider the mapsx ↦→ g ± (x) := d ∂Ω (x) ± 1 2 δ2 (x),where d ∂Ω is the distance function to ∂Ω. For some β 1 > 0 small, g ± are smooth inΩ β1and since |∇g ± | ≥ C > 0 on Σ k , by the implicit function theorem, the sets{x ∈ Ω β : g ± = 0}20

More magazines by this user
Similar magazines