Optimal rearrangement problem and normalized obstacle problem in the fractional setting

We consider an optimal rearrangement minimization problem involving the fractional Laplace operator $(-\Delta)^s$, $0<s<1$, and Gagliardo-Nirenberg seminorm $|u|_s$. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satisfies $$ -(-\Delta)^s U-\chi_{\{U\leq 0\}}\min\{-(-\Delta)^s U^+;1\}=\chi_{\{U>0\}}, $$ which happens to be the fractional analogue of the normalized obstacle problem $\Delta u=\chi_{\{u>0\}}$. A new section analyzing $s \to 1$ has been added.


Introduction
One of the classical problems in rearrangement theory is the minimization of the functional Recall thatR β is the closure in the weak* topology of the rearrangement class This minimization problem is related to the stationary heat equation Di erent force functions f result di erent heat distributions u f . The minimizerf of the functional (1.1) is the force function from a certain rearrangement class R, which is resulting the most uniformly distributed heat uf . 5 The problem and its variations, such as the p−harmonic case, has been studied by several authors (see [5-7, 13, 18]), and the results, for this particular setting, can be formulated in the following theorem. We refrain from presenting here details about the normalized obstacle problem (1.3), which is one of the 15 classical free boundary problems (see [9]). In recent years there has been a great development of nonlocal di usion problems, mainly due to some interesting new applications to di erent elds of the natural sciences such as some physical models [12,14,15,19,24,30], nance [2,20,27], uid dynamics [10], ecology [17,23,26] and image processing [16]. For a comprehensive reference for non-local di usion problems and their applications see [4]. 20 It is also worth mentioning that the link between non-local di usion problems and optimal design problems has been considered in recently in [22,29].
Among these models for nonlocal di usion, probably the most important one is given by the fractional laplacian (−∆) s , ( < s < ) that is given (for smooth functions) as This operator is given as the gradient of the nonlocal Gagliardo energy that is the nonlocal analog of the Dirichlet energy ∇u . In view of the increasing interest in analyzing nonlocal di usion models, it naturally comes into attention 25 considering problem (1.2) where the Laplace operator is replace by its fractional counterpart. Therefore, in this paper, similar to the way it has been done in [25], we will consider an optimal rearrangement problem and derive a related free boundary problem.
More precisely, we consider the minimization problem where We show existence and uniqueness of a solution to the fractional rearrangement optimization problem and show that iff is the solution andû = uf , then ≤û ≤ α for some α > and, moreover,Û = α −û is the unique solution to the normalized fractional obstacle problem Also, we analyze the behavior of such solutions as the fractional parameter s goes to 1.
Finally, we show that the solution to the fractional normalized obstacle problem is also the solution to the (highly nonlinear) equation

Organization of the paper
In Section 2 we give a brief introduction to fractional calculus, in Section 3 we analyze the optimal rearrange-5 ment problem in the fractional setting and show its relation with the normalized fractional obstacle problem. In Section 4, we study the behavior of the optimal fractional rearrangement problem as s → . Finally, in Section 5, we further analyze the normalized fractional obstacle problem and derive a (highly) nonlinear equation that the solution satis es.

. A very short tour through the basics of the fractional Laplacian
All the results in this section are either well-known or easily proved, so we just recall them for further references without any attempt of giving proofs.
The fractional order Sobolev spaces H s (R n ) (for < s < ) is de ned as where | · |s is the Gagliardo energy given by (1.4). This space is a Hilbert space with inner product given by For a brief summary of the properties of fractional order Sobolev spaces H s , we refer to the survey article [12]. Further we denote by H −s (R n ) the topological dual space of H s (R n ) and for a domain D ⊂ R n , we denote Recall that for Lipschitz domains D, the space H s (D) coincides with the closure of test functions with compact support inside D. We will also denote by H −s (D) the topological dual space of H s (D). Observe that we have Recall that if D is bounded, the following Poincaré type inequality holds true u ≤ C|u|s for all u ∈ H s (D). (2.1) An easy fact is that the Gagliardo semi-norm | · | s is Gâteaux -di erentiable in H s (R n ) and for every u, v ∈ H s (R n ). Furthermore, for a function u ∈ H s (R n ) we can also de ne the fractional Laplace operator as |x − y| n+ s dy and the limit is understood in H −s (R n ). 5 Moreover, it holds that The interested reader may consult with [1] for much more on the fractional laplacian and an analysis of di erences and similarities with the local Laplace operator.
For any f ∈ H −s (D) we say u f ∈ H s (D) solves the fractional boundary value problem in D with homogeneous Dirichlet boundary condition if the equation is satis ed in the sense of distributions. Equivalently, if for any v ∈ H s (D). It is easily seen from Riesz representation Theorem, using Poincaré inequality (2.1), that for any f ∈ H −s (D) there exists a unique u f ∈ H s (D) satisfying (2.5).
To nish these preliminaries we refer the reader to [28], and recall that for f ∈ L ∞ (D) the weak solution of (2.4),

The optimal fractional rearrangement problem
Let us now introduce the fractional analogue of the optimal rearrangement problem given in Theorem 1.1. Throughout this paper D ⊂ R n will always denote a bounded domain. Given f ∈ L (D), let u f be the solution 20 of (2.4) and let us de ne the functional We are going to consider the minimization of the functional Φs on the closed, convex setR β , for < β < |D|.
The main result of this section is the following theorem.
Remark 3.2. Observe that this result shows a remarkable di erence with the local optimal rearrangement problem, since the optimal con gurationf for the fractional case is not a characteristic function. See Theorem 1.1.
For the proof of Theorem 3.1 we need a couple of lemmas.
where for a convex set C, ext(C) denotes the extreme points of C. 5 Proof. The proof is standard and is omitted. For a more general result see [8,Lemma 3]. Proof. The strict convexity is a direct consequence of the linearity u f +f = u f + u f and the strict convexity of t → t . Moreover, from (2.5), Hölder's inequality and (2.1), we obtain Therefore, f → u f is strongly continuous from L (D) into H s (D) and hence, Φs is strongly continuous from 10 L (D) into R. Since Φs is convex, it follows that is sequentially weakly lower semicontinuous. Finally, observe that if fn ∈R β is such that fn * f weakly* in L ∞ (D), then fn f weakly in L (D), and so

Φs(f ) ≤ lim inf Φs(fn).
To nish the proof just notice that the existence of a minimizer follows from Banach-Alaoglu's theorem and the uniqueness of the minimizer from the strict convexity of Φs and the convexity ofR β . Now we are ready to prove Theorem 3.1.
Proof of Theorem 3.1. The proof will be divided into a series of claims.

Claim 2.
There exists a functionf = χ E ∈ R β such that The proof is an immediate consequence of the bathtub principle forf . See [21,Theorem 1.14].
The proof is again an immediate consequence of the bathtub principle forf .
Sincef ,f ∈R β , we have that and thus the claim.
Since (−∆) sû =f ∈ L ∞ (D) andf ≥ it is enough to checkf > point-wise. Taken Claim 4 we need to check this only in the set {û = α}. But This proves the claim.
The proof of Theorem 3.1 is complete.

5
The behavior of the optimal rearrangement problem as s → In this section we analyze the behavior of the optimal fractional rearrangement problem as the fractional parameter s goes to 1. For that purpose, we need to consider here the normalizing constant C(n, s) that is de ned as It is a well known fact that this normalizing constant behaves like ( −s) for s close to 1. Moreover, the following result holds where the rst limit is understood as a limit if u ∈ H (R n ) and as lim inf |u| s = ∞ if u ̸ ∈ H (R n ) and the second limit is in the sense of distributions. 10 For a proof, see for instance [12] and [3]. Moreover, it is shown in [3] the following stronger statement. Throughout this section, we will denote byfs the optimal load for Φs,ûs = uf s the solution to (2.4). Also, denote Φ(f ) as where in this section, u f will denote the solution to Finally, denote byf ∈R β the solution to the minimization problem So the main result in this section is the following: For the proof of Theorem 4.3 we need the concept of Γ−convergece. This concept was introduced by De Giorgi in the 60s and is now a well understood tool to deal with the convergence of minimum problems. For a throughout introduction to the subject, we cite [11]. Let us recall now the de nition of Γ−convergence and 5 some of its properties.

De nition 4.4.
Let X be a metric space and Fn , F : X →R. We say that Fn Γ−converges to F, and is denoted by Fn Γ → F, is the following two inequalities hold true • (lim inf −inequality) For any x ∈ X and any sequence {xn} n∈N ⊂ X such that xn → x in X, it holds that The main feature of the Γ−convergence is that it implies the convergence of minima. In fact we have the following: The proof of Theorem 4.5 is easy and can be found in [11].
The following result is key in the proof of Theorem 4.3. Since The trivial estimate |us|s ≤ fs imply that, for any s k → , the sequence {us k } k∈N ⊂ L (R n ) is precompact. Then from Theorem 4.5 we obtain that us → u strongly in L (Ω). Proof of Theorem 4.3. Letfs ∈R β be the optimal load for Φs. Observe that, for a subsequence,fs * f weakly* in L ∞ (Ω) for some f ∈R β . Moreover, this convergence also holds weakly in L (Ω). From Theorem 4.6 we have thatûs → u f strongly in L (Ω) and using Theorem 4.2 we get On the other hand, letf ∈R β be the optimal load for Φ. Then, using the nal part of Theorem 4.6, we obtain lim sup inf The proof is complete.

The normalized fractional obstacle problem
This section is devoted to the study of the connection between the solutions to the optimal fractional rearrangement problem consider in Section 3 with solutions of the normalized fractional obstacle problem.
The fractional analogue of the classical obstacle problem has been well known in the literature, however its so called normalized version, i.e., the equation has not been considered. Here we nd the corresponding fractional analog of (5.1) and prove that the solution of the fractional rearrangement problem is a solution of the fractional normalized obstacle problem. Our rst result is the following.
in the sense of distributions. 10 Finally, the minimizer of J in Hα is unique and is the unique solution to the inequality (5.2).

Proof. Let
and observe that, since ≤f ≤ , for any v ∈ Hα it follows that J(v) ≥ I(v). Next, observe that I(Û) = J(Û) and so the set of inequalities imply the desired result. Next, observe that the inequalities are the Euler-Lagrange equation for the functional J based on the variation uε(x) = u(x) + εϕ(x), with ϕ ∈ C ∞ c (D). 15 Now, the uniqueness of minimizer for J is an immediate consequence of the strict convexity of J. Assume that the function U satis es the inequalities (5.2), but the unique minimizer of the convex functional J is the function V ≠ U.
Since J is strictly convex and J(V) < J(U) by taking Uε = U + ε(V − U) we will obtain where the last inequality follows from (5.2). This is a contradiction and the result follows.
Remark 5.2. This result again shows an interesting di erence between the classical obstacle problem and the fractional normalized version. Observe that in the positivity set, we still have −(−∆) sÛ = , but in the zero set the functionÛ is not s−harmonic (even if it is identically zero!). The free boundary condition on ∂{Û > } is given by the fact that (−∆) sÛ is a function bounded by and across the free boundary. 5 The results in Theorem 5.1 are not completely satisfactory, since we do not obtain an equation satis ed byÛ but only the inequalities (5.2).
Our last result shows that in factÛ is the solution to a fully nonlinear equation. Proof. If u is smooth, then the fractional laplacian has pointwise values. In this case, we simply compute: For a general u ∈ H s (R n ), we take {ρε} ε> a smooth family of approximations of the identity such that ρε(z) = ρε(−z) and de ne uε = u * ρε.