Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods

Let u : R → R be nonnegative, 2 ⁢ π -periodic, and such that

Let E = { ( x 1 , x 2 ) ∈ R 2 : | x 2 | < u ⁢ ( x 1 ) } . Then (A.1) holds.

Proof

We prove (A.1) using (2.2). Let us first rewrite 𝜙 in (2.3) for n = 2 in a more convenient way by introducing the function G ′ . For p ≥ 0 and q ≥ 0 , we have (using the change of variables z = τ + w twice)

Note that G ′ is odd, 0 < − G ′ ⁢ ( − ∞ ) = G ′ ⁢ ( + ∞ ) < + ∞ , and G ′ ( − q − ⋅ ) − G ′ ( q − ⋅ ) is an even function. Therefore,

where we also used in the last equality that 𝐺 is even. Now, from (2.2) and (A.2), we deduce that

Our goal is to show that, in the term involving 2 ⁢ G ′ ⁢ ( + ∞ ) ⁢ u ⁢ ( x 1 ) ⁢ | x 1 − y 1 | − 1 , we can replace u ⁢ ( x 1 ) by u ⁢ ( y 1 ) and identity (A.3) remains unchanged. Once this is proved, we get that, in this last term, one can replace 2 ⁢ u ⁢ ( x 1 ) by u ⁢ ( x 1 ) + u ⁢ ( y 1 ) . Hence, this gives (A.1), using that H ⁢ ( t ) = G ′ ⁢ ( + ∞ ) ⁢ t − G ⁢ ( t ) .

Therefore, it only remains to show that we can replace u ⁢ ( x 1 ) by u ⁢ ( y 1 ) in the last term on the right-hand side of (A.3). This follows immediately from Lemma 2.1 (used with m = l = 1 ) applied to the integral

where

To check the hypothesis of the lemma, we need to verify that

For this purpose, we write f ⁢ ( x , y ) = A + B , where

Using that 𝑢 is nonnegative, 𝐻 is nonnegative and increasing in ( 0 , + ∞ ) , and H ⁢ ( + ∞ ) < + ∞ , we have

Concerning the quantity 𝐴, observe that

Recall that we are assuming u ∈ W s , 1 ⁢ ( − π − ϵ , π + ϵ ) for some ϵ > 0 . According to this, to prove (A.4), we split the integral and we use (A.5) and (A.6) to get

The first two terms on the right-hand side of (A.7) are finite. To estimate the third one, we use that 0 ≤ G ⁢ ( t ) ≤ G ′ ⁢ ( + ∞ ) ⁢ t for all t ≥ 0 , and hence

The first double integral in the right-hand side of (A.8) is finite since s > 0 and

For the second double integral, since 𝑢 is 2 ⁢ π -periodic and nonnegative, using Lemma 2.1, we deduce

As before, this is finite since s > 0 and u ∈ L 1 ⁢ ( − π , π ) . Hence (A.4) follows from (A.7) and the finiteness of the integrals in (A.8). ∎

For every t > 0 , the function z ↦ Γ ⁢ ( z , t ) is decreasing in ( 0 , π ) .

Proof

Let us first show that Γ ⁢ ( ⋅ , t ) is nonincreasing in ( 0 , π ) . For this, we take functions g ϵ ∈ C c ∞ ⁢ ( − ϵ , ϵ ) , where 0 < ϵ < π / 2 , approximating the Dirac delta as ϵ ↓ 0 , and we consider their 2 ⁢ π -periodic extensions from [ − π , π ] to ℝ (which we also denote by g ϵ ). In particular, we assume that

We additionally assume that the g ϵ are nonnegative, even, and nonincreasing in ( 0 , π ) .

Let u ϵ be the solution to

Thus, u ϵ ⁢ ( x , t ) = ∫ − π π d y ⁢ Γ ⁢ ( x − y , t ) ⁢ g ϵ ⁢ ( y ) and u ϵ ⁢ ( ⋅ , t ) is 2 ⁢ π -periodic for all t > 0 . Notice that u ϵ ⁢ ( ⋅ , t ) is even with respect to x = 0 and x = π (by uniqueness, since so is g ϵ ). We deduce that the derivative v ϵ := ∂ x u ϵ solves

Moreover, by the assumptions on g ϵ , we have that v ϵ ⁢ ( x , 0 ) = ∂ x u ϵ ⁢ ( x , 0 ) = g ϵ ′ ⁢ ( x ) ≤ 0 for all x ∈ [ 0 , π ] . Hence, the maximum principle yields

We now claim that, for every given t > 0 and m = 0 , 1 , 2 , … ,

This follows by combining the uniform continuity in [ 0 , π ] of Γ ⁢ ( ⋅ , t ) and of all its derivatives with the fact that, for every δ > 0 , we have

if 𝜖 is chosen small enough.

Now, from (B.1) and (B.2), we conclude that

for all x ∈ [ 0 , π ] and t > 0 .

Finally, we show that Γ ⁢ ( ⋅ , t ) is decreasing in ( 0 , π ) . For this, notice that we already proved that, for all t > 0 , Γ ⁢ ( ⋅ , t ) is even with respect to x = 0 and x = π . Therefore, given any t 0 > 0 , we see that v := ∂ x Γ solves

Now, the strong maximum principle yields that either v < 0 in ( 0 , π ) × ( t 0 , + ∞ ) or v ≡ 0 in [ 0 , π ] × [ t 0 , + ∞ ) . The proof is now complete since v ≡ 0 is absurd – it would give that Γ ⁢ ( ⋅ , t ) is constant in 𝑥, clearly contradicting the fact that Γ is the fundamental solution of the heat equation with periodic boundary conditions, as shown in the beginning of this appendix. ∎