Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains

Proposition A.1.

Under the assumptions of Theorem 2.4, except for Ω, formula (2.5) holds if:

1. Ω is of class C 2 , σ , for any σ ∈ ( 0 , 1 ) , where the term ∫ ∂ ⁡ Ω ℋ ⁢ | ∇ ⁡ U | 2 ⁢ | V | 2 is understood with ℋ the curvature in the classical sense, or

2. Ω is convex, and in this case the term with ℋ is understood as

   (A.1) ∫ ∂ ⁡ Ω ℋ ⁢ | ∇ ⁡ U | 2 ⁢ | V | 2 := - ∫ Ω [ ∂ j ⁡ ∂ i ⊥ ⁡ U ] ⁢ ∂ i ⁡ ( ∂ j ⊥ ⁡ U ⁢ | V | 2 ) = lim ε → 0 ⁡ ∫ ∂ ⁡ Ω ε ℋ ε ⁢ | ∇ ⁡ U | 2 ⁢ | V | 2 ⁢ 𝑑 s ε .

Similarly, under the assumptions of Theorem 2.7, the same statements as above hold for λ 1 ′′ ⁢ ( Ω ) ⁢ ( V , V ) given by

Proof.

First we prove the result for E f ′′ ⁢ ( Ω ) . In both (i) and (ii) cases we follow the proof of Theorem 2.4.

In case (i), we can repeat all the calculus of Theorem 2.4 with Ω ε replaced by Ω (and so, without the need to consider the limits of different terms as ε tends to zero).

In case (ii), we need to identify lim ε → 0 ⁡ K 3 ⁢ ( ε ) . For this we use the following extensions for the normal and tangential vectors on ∂ ⁡ Ω ε , ν ε = - ∇ ⁡ U | ∇ ⁡ U | , τ ε = ν ε ⊥ = - ∇ ⊥ ⁡ U | ∇ ⁡ U | . Then we have

The proof of the result for λ 1 ′′ ⁢ ( Ω ) is similar to the proof for E f ′′ ⁢ ( Ω ) and we do not present it here. ∎

Remark A.2.

In the case Ω is convex, ℋ ⁢ | ∇ ⁡ U | 2 is not well defined because, for example, ℋ is unbounded and | ∇ ⁡ U | is zero at a corner. However, as the calculus proceeding (A.2) holds with | V | 2 = w ¯ , for every w ∈ W 1 , ∞ ⁢ ( ∂ ⁡ Ω ) and w ¯ the extension of w given by Lemma 1.12, it implies that (A.1) holds with | V | 2 = w ¯ . This defines ℋ ⁢ | ∇ ⁡ U | 2 as an element of ( W 1 , ∞ ⁢ ( ∂ ⁡ Ω ) ) ′ , the dual space of W 1 , ∞ ⁢ ( ∂ ⁡ Ω ) .

Corollary A.3.

For v = ( v 1 , v 2 ) ∈ W 1 , ∞ ⁢ ( ∂ ⁡ Ω ; R 2 ) let V = ( v ¯ 1 , v ¯ 2 ) be the extension of v as given by Lemma 1.12. Then, under the same assumptions as in theorem 2.4 we have:

1. The map v ∈ W 1 , ∞ ⁢ ( ∂ ⁡ Ω ; ℝ 2 ) ↦ E f ′ ⁢ ( Ω ) ⁢ ( V ) extends continuously in L 1 ⁢ ( ∂ ⁡ Ω ; ℝ 2 ) and the extension is given by ( 2.7 ).

2. Similarly, the map v ∈ W 1 , ∞ ⁢ ( ∂ ⁡ Ω ; ℝ 2 ) ↦ E f ′′ ⁢ ( Ω ) ⁢ ( V , V ) extends continuously in H 1 ⁢ ( ∂ ⁡ Ω ; ℝ 2 ) ∩ L ∞ ⁢ ( ∂ ⁡ Ω ; ℝ 2 ) and the extension is given by ( 2.5 ).

Proof.

First we note that the extension H of Lemma 1.12 extends continuously in L 1 ⁢ ( ∂ ⁡ Ω ) and H 1 ⁢ ( ∂ ⁡ Ω ) . Then for (i) we note that (2.7) is continuous with respect to V ∈ L 1 ⁢ ( ∂ ⁡ Ω ; ℝ 2 ) , so the claim follows from the continuity of H in L 1 . For claim (ii), first we note that all the terms of (2.5) except ∫ ∂ ⁡ Ω ℋ ⁢ ( ∂ ν ⁡ U ) 2 ⁢ | V | 2 are continuous in H 1 ⁢ ( ∂ ⁡ Ω ) with respect to V, so they are continuous with respect to v in H 1 ⁢ ( ∂ ⁡ Ω ; ℝ 2 ) thanks to the continuity of H in H 1 . For the term ∫ ∂ ⁡ Ω ℋ ⁢ ( ∂ ν ⁡ U ) 2 ⁢ | V | 2 , from (A.1) with w ¯ = | V | 2 we see that this term is continuous with respect to V in H 1 ⁢ ( Ω ; ℝ 2 ) ∩ L ∞ ⁢ ( Ω ; ℝ 2 ) and we conclude using again the continuity of H in H 1 ∩ L ∞ . ∎

Lemma A.4.

The equivalence of seminorms (4.10) holds.

Proof.

It is classical, see for example [7], that for h ∈ H 1 / 2 ⁢ ( ∂ ⁡ Ω ) , its seminorm is given by

For θ , η ∈ ( - π , π ) , x = ( cos ⁡ θ , sin ⁡ θ ) , y = ( cos ⁡ η , sin ⁡ η ) we set (with a slight abuse of notations) h ⁢ ( θ ) = h ⁢ ( x ) , h ⁢ ( η ) = h ⁢ ( y ) . Then (A.3) is equal to

where

and

Let θ ^ = θ - 2 ⁢ π , η ^ = η , A ^ 2 = A 2 - ( 2 ⁢ π , 0 ) . Then

One can check that

Similarly, let θ ^ = θ + 2 ⁢ π , η ^ = η , A ^ 3 = A 3 + ( 2 ⁢ π , 0 ) . Then

Similarly to A ^ 2 , one can check that

Therefore we get

because A 1 ∪ A ^ 2 ∪ A ^ 3 is the parallelogram { ( θ , η ) : η ∈ ( - π , π ) , | θ - η | < π } . Note that we have

which combined with the last equality proves the claim. ∎