A Weierstrass extremal field theory for the fractional Laplacian

A A calibration for the extension problem of the fractional Laplacian

In this appendix, we study minimizers of the energy functional ℰ s , F by using the extension technique for the fractional Laplacian. The strategy is based on building a calibration for an auxiliary local energy ℰ ~ s , F in the extended space ℝ + n + 1 = ℝ n × ( 0 , ∞ ) . We point out that this construction did not give us, during the conception of our work, any a priori information about the form of a calibration written “downstairs” (i.e., in ℝ n ) for the original energy functional ℰ s , F . It was only after finding 𝒞 s , F by nonlocal arguments that we noticed how to deduce it, at least formally, from the extension problem.

We denote by ( x , z ) ∈ ℝ n × ℝ points in ℝ + n + 1 . Given a bounded domain Ω ⊂ ℝ n , we say that a bounded set Ω ~ ⊂ ℝ + n + 1 is an extension of Ω if

It is well known that there is a strong connection between the nonlocal energy functional ℰ s , F and the local one

where d s is a positive normalizing constant. For this, given a function u defined in ℝ n , we consider U being the solution of

Here, U is the so-called s-harmonic extension of u. In [10, Lemma 7.2], Caffarelli, Roquejoffre, and Savin showed that u is a minimizer of ℰ s , F among functions with the same exterior data as u in Ω c if and only if, for every extension domain Ω ~ , the s-harmonic extension U of u is a minimizer of ℰ ~ s ⁢ ( ⋅ ; Ω ~ ) among functions with the same boundary condition as U on

Taking into account this equivalence, we can apply the classical theory of calibrations to the mixed Dirichlet–Neumann problem as explained in Remark 3.6. To do this, given a field { u t } t ∈ I in ℝ n , for some interval I ⊂ ℝ , it is clear by the maximum principle that we can define a new field { U t } t ∈ I in ℝ + n + 1 where each leaf U t is the s-harmonic extension of u t . Then the functional

can be proved to be a calibration for ℰ ~ s , F and U. Therefore, U is a minimizer of ℰ ~ s , F , and by [10, Lemma 7.2] it follows that u is a minimizer of ℰ s , F .

We point out that, although in this way we easily found a calibration for the local energy ℰ ~ s , F ⁢ ( ⋅ ; Ω ~ ) , it was not clear at all how it translated into a calibration written “downstairs” for the original energy functional ℰ s , F . It was only after building the calibration 𝒞 s , F by using purely nonlocal techniques that we discovered how to pass, at least formally, from 𝒞 ~ s , F ⁢ ( ⋅ ; Ω ~ ) to 𝒞 s , F . Letus explain this. First, as in Section 3, for t 0 ∈ I , we rewrite (A.1) in the alternative form^[18]

where ν ∂ L ⁡ Ω ~ is the exterior normal vector to the lateral boundary ∂ L ⁡ Ω ~ . Notice here that in the second term we have used the identity

which follows from the Caffarelli–Silvestre extension; see [11]. Eventually, taking a sequence of extended domains Ω ~ i converging to the half-space ℝ + n + 1 , we recover the functional 𝒞 s , F (up to an additive constant) as the formal limit of 𝒞 ~ s , F ⁢ ( ⋅ ; Ω ~ i ) .

B Other candidates for the fractional calibration

In this appendix, we discuss three other natural candidates to be a calibration for the energy functional

We will be able to discard two of them since some of the calibration properties fail in these cases. Nevertheless, there is still one candidate for which we cannot determine whether it is a calibration or not.

Let us recall that the local counterpart of ℰ s , F is the functional

which admits the calibration

a functional that can also be written as

Inspired by the form of 𝒞 1 , F , the first natural calibration candidate for ℰ s , F can be built replacing the gradient terms by differences and double integrals. That is, we let

By using Young’s inequality and the definition of the leaf-parameter function, one can directly conclude that ℱ s , F 1 satisfies properties (C1) and (C2). It remains to check whether the null-Lagrangian property (C3) is satisfied, but we do not know how to answer this question. For the affirmative answer, the idea would be to use the usual nonlocal integration by parts technique to obtain the Euler–Lagrange equation on the leaves. However, since the leaf-parameter function t depends on the variable x, we get remainder terms that we do not know how to treat. It is then natural to look for a counterexample. We looked at cases where an explicit field is available. For the trivial potential F = 0 , for which u t ⁢ ( x ) = x + t are extremals (even if not bounded), property (C3) does not fail. Hence, this case does not discard the candidate ℱ s , F 1 . Another interesting example with explicit solutions is the Peierls–Nabarro model, corresponding to the case n = 1 , s = 1 2 , and F ⁢ ( u ) = 1 - cos ⁡ ( u ) . Here the equation ( - Δ ) 1 / 2 ⁢ u = sin ⁡ ( u ) in ℝ admits the field of extremals u t ⁢ ( x ) = 2 ⁢ arctan ⁡ ( x + t ) . We do not know if the null-Lagrangian property holds for ℱ s , F 1 in this concrete example.

It is also interesting to compare ℱ s , F 1 with the calibration 𝒞 s , F constructed in Section 4. There, by the alternative expression for 𝒞 s , F derived in Lemma 4.4, we see that ℱ s , F 1 ⁢ ( w ) and 𝒞 s , F ⁢ ( w ) would coincide if the following equality were true:

However, we do not know how to prove or disprove this identity.

The functional ℱ s , F 1 does not capture the symmetry in the variables x and y that has appeared in the two previous works on nonlocal calibrations [5, 22]. Hence, it is also natural to propose the following new candidate:

As in the preceding case, we can apply Young’s inequality and the definition of the leaf-parameter function to deduce that ℱ s , F 2 satisfies properties (C1) and (C2). Nevertheless, in this case we can discard it as a calibration since the null-Lagrangian property fails even when F = 0 and u t ⁢ ( x ) = x + t .

One could also think of a calibration candidate constructed by replacing the gradient terms in the local theory by fractional ones. That is,

Here, the fractional gradient is defined by

This last candidate would be motivated by the identity

Nevertheless, a similar equality does not hold when restricting to a domain Ω, i.e.,

Hence, ℱ s , F 3 does not satisfy property (C1) and thus it is not a calibration for ℰ s , F .