The sharp quantitative isocapacitary inequality (the case of p-capacity)

Lemma A.1 ([14, Lemma 2.3]).

Let p > 1 . There exists c ⁢ ( p ) ≥ 0 such that if κ ≥ 0 and ξ , η ∈ R n , then

Moreover, there exists another constant C ⁢ ( p ) ≥ 0 such that if Ω ⊂ R n is an open set and for u , v ∈ W 1 , p ⁢ ( Ω ) and 0 ≤ s ≤ 1 , we set u s ⁢ ( x ) = s ⁢ u ⁢ ( x ) + ( 1 - s ) ⁢ v ⁢ ( x ) , then the following two inequalities hold:

1. for p ≥ 2 ,

   (A.1) ∫ Ω | ∇ ⁡ u - ∇ ⁡ v | p ≤ C ⁢ ∫ 0 1 1 s ⁢ 𝑑 s ⁢ ∫ Ω ( | ∇ ⁡ u s | p - 2 ⁢ ∇ ⁡ u s - | ∇ ⁡ v | p - 2 ⁢ ∇ ⁡ v ) ⋅ ∇ ⁡ ( u s - v ) ,

2. for 1 < p < 2 ,

   (A.2) ∫ Ω | ∇ ⁡ u - ∇ ⁡ v | p ≤ C ⁢ ( ∫ 0 1 1 s ⁢ 𝑑 s ⁢ ∫ Ω ( | ∇ ⁡ u s | p - 2 ⁢ ∇ ⁡ u s - | ∇ ⁡ v | p - 2 ⁢ ∇ ⁡ v ) ⋅ ∇ ⁡ ( u s - v ) ) p 2 ⁢ ( ∫ Ω ( | ∇ ⁡ u | + | ∇ ⁡ v | ) p ) 1 - p 2 .

Lemma A.2.

Let x , y ∈ R N , p ∈ ( 1 , ∞ ) . Then the following inequalities hold:

1. if p ≥ 2 , then

   | y | p ≥ | x | p + p ⁢ | x | p - 2 ⁢ x ⋅ ( y - x ) + c ⁢ | y - x | p

   for some c = c ⁢ ( p ) > 0 ,

2. if 1 < p < 2 , then

   | y | p ≥ | x | p + p ⁢ | x | p - 2 ⁢ x ⋅ ( y - x ) + c ⁢ | y - x | 2 ⁢ ( | x | 2 + | y - x | 2 ) p - 2 2

   for some c = c ⁢ ( p ) > 0 .

Proof.

Consider a function f : ℝ N → ℝ defined as f ⁢ ( x ) = | x | p . Writing Taylor expansion for f, we get

Case p = 2 . We have

which gives us a desired inequality. We shall consider p ≠ 2 from now on.

Case p ≠ 2 . The Hessian D 2 ⁢ f ⁢ ( x ) looks as follows:

where A i , j = x i ⁢ x j . We notice that

yielding

where c = c ⁢ ( p ) > 0 ( c = p for p > 2 , c = p ⁢ ( p - 1 ) for 1 < p < 2 ). So, we have

Let us consider the cases of different p separately.

First, we deal with 1 < p < 2 . In this case p - 2 < 0 and so

finishing the proof of lemma in this case.

To tackle the case p > 2 , we further consider two cases. If | y - x | < 2 ⁢ | x | , then

If instead | y - x | ≥ 2 ⁢ | x | , then