On a fractional harmonic replacement

Given s ∈ (0, 1), we consider the problem of minimizing the Gagliardo seminorm in H with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set K . We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set A to K increases the energy of at most the measure of A (this may be seen as a perturbation result for small sets A). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.


INTRODUCTION
This paper deals with an harmonic replacement of nonlocal type with a prescribed zero set. We obtain energy monotonicity results with respect to the data and the zero set, and some perturbative estimates with respect to the variation of the zero set.
Next purpose of our paper is to estimate the energy difference of the s-harmonic replacements of K and K ∪ A, for a given set A (which can be seen as a "perturbation") in terms of the Lebesgue measure of A. The results that we provide are the following: , for some C > 0 depending on n and s. Theorem 1.4. Let ϕ 0, K ⊇ B 1/2 and A ⊆ B 3/4 \ B 1/2 . Suppose that A is closed and satisfies the following density property: there exists c > 0 such that for every x ∈ ∂A and every r ∈ (0, 2), we have that (1.6) |A ∩ B r (x)| c |B r |. Then ν 2 (ϕ K∪A ) − ν 2 (ϕ K ) C |A| ϕ K 2 L ∞ (R n ) , for some C > 0 depending on c, n and s.
We observe that sets with Lipschitz boundary obviously satisfy the density property in (1.6). Also, we notice that the geometry of the perturbing set A in Theorem 1.4 is different from the one in Theorem 1.3: namely, in Theorem 1.3 the set A may be thought as "exiting" from K in the interior of B 1 , while in Theorem 1.4 the set A "stretches out" from K towards the boundary. Possible pictures for the geometries of the sets involved in Theorems 1.3 and 1.4 are depicted in Figures A and B respectively.
In both the figures the set K is painted in black and A is the dark gray region (of course, Theorems 1.3 and 1.4 are interesting when A is a "small" perturbation, but for obvious aesthetic reasons the sets A drawn in the figures are "not so small").  In the local case of the harmonic replacement (i.e. the classical minimization problem of the Dirichlet energy) the results presented in this paper were obtained in [2]. Theorems 1.3 and 1.4 may be seen as perturbative statements, namely they estimate the change of energy in terms of the (possibly small) set A. It is worth pointing out that the estimates obtained are simply in terms of the Lebesgue measure of A and only require very mild regularity assumptions on the set (in fact, only the density assumption (1.6), and no high derivative of the boundary of A comes into play).
We also observe that once Theorems 1.3 and 1.4 are proved for minimizers in a ball (say, B 1 ), then they hold true for minimizers in any open set Ω: this follows from the fact that one can suppose Ω ⊃ B 1 (up to scaling) and obtain from the set inclusions that |ϕ(x) − ϕ(y)| 2 |x − y| n+2s dx dy and the latter two integral terms do not depend on the values of ϕ in B 1 : accordingly, a minimizer in a domain Ω which contains B 1 is also a minimizer in B 1 .
Also, the values 1/4, 1/2 and 3/4 in Theorems 1.3 and 1.4 do not play any role, in the sense that they can be replaced by some r 1 , r 2 and r 3 respectively, with 0 < r 1 < r 2 < r 3 < 1 (but in this case the constants would depend on r 1 , r 2 and r 3 ).
As for the applications of our result, we notice that, in the local setting, the Dirichlet integral may be interpreted in terms of the classical heat equation as a sort of thermal energy: in this sense, the Dirichlet integral of the harmonic replacement with boundary data ϕ 0 that vanishes in K represents the insulating energy of a room whose walls are fixed at temperature ϕ and having a "fridge" at the set K where the temperature is zero. In this framework, we may consider the fractional harmonic replacement as a nonlocal modification of this problem, in which the classical heat equation is replaced by a nonlocal one, which is generated by a non-Gaussian diffusive process, see e.g. [9]. Also, harmonic replacements play an important role in the Perron method and in free boundary problems, see e.g. [5] and [1].
The paper is organized as follows. In Section 2 we show some properties of the fractional harmonic replacement. In Section 3 we deal with the monotonicity property given in Theorem 1. Proof. First we prove the existence of the minimum. For this, let v j ∈ D ϕ K be a minimizing sequence.
In particular, from (1.1), we may suppose that ν(v j ) ν(v), which is finite. Set w j := v j − ϕ. Then w j vanishes outside B 1 , thus which is finite, thanks to (1.1). Therefore (see, e.g. Theorem 7.1 in [3]) we obtain, up to subsequence, that w j converges in L 2 (B 1 ) and a.e. in R n to some w. Accordingly, v j converges in L 2 (B 1 ) and a.e.
in R n to v := w + ϕ. In particular, v ∈ L 2 (B 1 ), v = 0 a.e. in K and v = ϕ a.e. in B c 1 . Moreover, by Fatou Lemma, which says that ν(v) is finite. Thus v ∈ D ϕ K and attains the desired minimum. Now we show that the minimizer is unique. Suppose that u and v are minimizers in D ϕ We denote δu(x, y) := u(x) − u(y). For any r ∈ R n , let also f (r) := |r| 2 . By convexity strict inequality in (2.2) holds whenever δu(x, y) = δv(x, y).
Indeed, assume by contradiction that Z has positive measure. By dividing by |x − y| n+2s and integrating (2.2), and recalling (2.1), we see that This contradiction establishes (2.4).
By construction, we have that equality holds in (2.2) for every (x, y) ∈ R 2n \Z, and therefore, by (2.3), Now we observe that there existȳ ∈ R n and V ⊂ R n such that |V| = 0, and (x,ȳ) ∈ R 2n \ Z for any x ∈ R n \ V.
(2.6) 6 The proof of (2.6) relies on Fubini's theorem, we give the details for completeness. For any y ∈ R n , Then b is a nonnegative and measurable function, and due to (2.4). Accordingly b(y) = 0 for a.e. y ∈ R n . In particular, we can fixȳ ∈ R n such that b(ȳ) = 0, that is As a consequence χ Z (x,ȳ) = 0 for a.e. x ∈ R n (say, for every x ∈ R n \ V, for a suitable V ⊂ R n of zero measure). This concludes the proof of (2.6). By , we obtain that c = 0, and therefore u = v a.e. in R n . This completes the uniqueness result and ends the proof of Lemma 2.1.
Additional properties of the s-harmonic replacement hold true if the datum ϕ has a sign, according to the next results.
Proof. By Lemma 2.5, (2.15) and (2.8), we have that Viceversa, using (2.8) and (2.16), we have that By combining the two inequalities, we obtain the desired result.

MONOTONICITY PROPERTY AND PROOF OF THEOREM 1.2
In the light of the lemmata discussed in Section 2, we can now prove the monotonicity property of s-harmonic replacements: Proof of Theorem 1.2. We letv minimize ν among all the functions v such that v −φ 2 ∈ H s (R n ), with v =φ 2 a.e. in K 1 ∪B c 1 . Notice thatφ 2 is an admissible competitor for this definition, and ν(φ 2 ) < +∞, hence the minimum definingv is attained by direct methods (see Lemma 2.1).
From this and recalling (2.10) with u := ϕ 1 −v, we obtain that But the latter term also vanish, thanks to (2.7): therefore we conclude that ν 2 (h) 0 and so h vanishes identically.

RADIAL ANALYSIS AND PROOF OF THEOREM 1.3
This section is devoted to the proof of Theorem 1.3. First we prove Theorem 1.3 in the particular case in which ϕ is constant, K := B r and A := B ρ \ B r , for some r < ρ, namely we have the following result: , for some C > 0 depending on n and s.
Proof. If c o = 0 then both ϕ Bρ and ϕ Br vanish identically and the result is obvious. Hence, possibly dividing by c o , we suppose that (4.2) c o = 1.
On the other hand, when µ c, we can prove (4.1) directly from the competitor ϕ o introduced above. More precisely, if µ c, we infer from (4.5) that . This and (4.3) say that (4.1) holds true also when µ c and this completes the proof of Lemma 4.1.
Lemma 4.1 may be generalized to sets that are not necessarily rotationally symmetric, thanks to a rearrangement argument. The details go as follows: for some C > 0 depending on n and s.
Proof. We take r 0 such that |B r | = |K|. Let also ψ := c o − ϕ K . Then ψ = 0 in B c 1 and ψ = c o = max R n ψ in K, thanks to Lemma 2.4. Then, its spherical rearrangement ψ satisfies ψ = 0 in B c 1 and ψ = c o in B r . Accordingly, c o − ψ is a competitor against ϕ Br and so On the other hand, spherical rearrangements decrease the Gagliardo seminorm (see e.g. [6]), therefore We obtain that Notice also that K ∪ A = B ρ , hence, using Lemmata 4.1 and 2.4, we obtain that Now we are ready for the proof of Theorem 1.3: Proof of Theorem 1.3. We set ϕ := ϕ L ∞ (R n ) , K := K ∩ B ρ and A := B ρ \ K . We are now under the assumptions of Corollary 4.2, which gives that

By construction
Also, ϕ ϕ and K ⊆ K: therefore, by Theorem 1.2, By collecting the above estimates, and recalling Lemma 2.4, we complete the proof of Theorem 1.3.

INTEGRAL CALCULATIONS AND PROOF OF THEOREM 1.4
Here we prove Theorem 1.4 in the particular case in which K := B 1/2 and ϕ is constant (the general case then will follow from Theorem 1.2).
Assume that ϕ(x) = c o for any x ∈ R n and that A is closed and satisfies the following density property: there exists c > 0 such that for every x ∈ ∂A and every r ∈ (0, 2), we have that Proof. Notice that if ϕ B 1/2 L ∞ (R n ) = 0, then both ϕ B 1/2 and ϕ B 1/2 ∪A vanish identically and so the result is obvious. Hence, without loss of generality, we assume that ϕ B 1/2 L ∞ (R n ) = 1/4. We claim that (5.2) ϕ B 1/2 ∈ C s (R n ), and ϕ B 1/2 C s (R n ) is bounded by a constant that depends only on n and s. For this we take η ∈ C ∞ (R n ) such that η = 0 in B 1/2 and η = 1/4 in B c 1 . We defineη Accordingly, v(x) = ϕ B 1/2 (x) in B c 1 and v = 0 in B 1/2 ∪ A, so it is an admissible competitor for ϕ B 1/2 ∪A , and we conclude that To prove (5.5) we distinguish three cases: either y ∈ {ϕ B 1/2 < d} and ϕ B 1/2 (y) < d(x), (5.6) or y ∈ {ϕ B 1/2 < d} and ϕ B 1/2 (y) d(x), (5.7) or y ∈ {ϕ B 1/2 d}.
First we deal with the case in (5.6). For this, we notice that, in this circumstance, ϕ B 1/2 (x) > d(x) > ϕ B 1/2 (y) and so and so (5.5) follows in this case.
Now we deal with the case in (5.7). Here we have that d(y and so (5.5) follows in this case.
Hence, by the Vitali covering theorem, we have that there exists a subcollection of disjoint balls such that S ⊆ i∈N B δ(x i ) (y(x i )) and so (5.13) |S| i∈N |B δ(x i ) (y(x i ))|.
Using (5.1), we have that |A ∩ B δ(x i ) (y(x i ))| c |B δ(x i ) (y(x i ))|. So we can fix N ∈ N, sum up this estimate and use that the balls are disjoint: we obtain that |B δ(x i ) (y(x i ))|.
Now we send N → +∞ and we recall (5.13), to establish that |A| c |S|.
With the above results, we can now complete the proof of Theorem 1.4: Proof of Theorem 1.4. We set ϕ := ϕ L ∞ (R n ) and K := B 1/2 . We are now under the assumptions of Lemma 5.1 which gives that ν 2 (ϕ K ∪A ) − ν 2 (ϕ K ) C |A| ϕ 2 L ∞ (R n ) . Notice also that K ⊆ K and ϕ ϕ, thus Theorem 1.2 implies that so the claim of Theorem 1.4 readily follows.