Regularity for the planar optimal p-compliance problem

In this paper we prove a partial $C^{1,\alpha}$ regularity result in dimension $N=2$ for the optimal $p$-compliance problem, extending for $p\not = 2$ some of the results obtained by A. Chambolle, J. Lamboley, A. Lemenant, E. Stepanov (2017). Because of the lack of good monotonicity estimates for the $p$-energy when $p\not = 2$, we employ an alternative technique based on a compactness argument leading to a $p$-energy decay at any flat point. We finally obtain that every optimal set has no loop, is Ahlfors regular, and $C^{1,\alpha}$ at $\mathcal{H}^1$-a.e. point for every $p \in (1 ,+\infty)$.


Introduction
Given an open bounded set Ω ⊂ R 2 , an exponent p ∈ (1, +∞) and a function f ∈ L p ′ (Ω) with p ′ = p/(p − 1), let Σ ⊂ Ω be a closed set and for u ∈ W 1,p 0 (Ω \ Σ) define It is classical that the functional E p admits a unique minimizer u Σ , which is the solution of the Dirichlet problem in the weak sense, which means that for all ϕ ∈ W 1,p 0 (Ω \ Σ). Following [6], we can interpret Ω as a membrane which is attached along Σ ∪∂ Ω to some fixed base (where Σ can be interpreted as the "glue line") and subjected to a given force f . Then u Σ is the displacement of the membrane. The rigidity of the membrane is measured through the p-compliance functional, which is defined as We study the following shape optimization problem. The physical interpretation of this problem may be the following: we are trying to find the best location Σ for the glue to put on the membrane Ω in order to maximize the rigidity of the latter, subject to the force f , while the penalization by λH 1 takes into account the quantity (or cost) of the glue.
Without loss of generality, we assume that the force field f is nonzero, because otherwise for any Σ ∈ K(Ω) we would have C p (Σ) = 0 and then every point x ∈ Ω becomes a solution of Problem 1.1.
In this paper we prove some regularity properties about the minimizers of Problem 1.1. In particular, we prove that a minimizer has no loop (Theorem 6.1) and is Ahlfors regular (Theorem 3.3). Furthermore, we establish some C 1,α regularity properties about the minimizers, as stated in the following main result.
Moreover, if Ω is a C 1 convex domain, then for H 1 -a.e. point x ∈ Σ one can find r 0 > 0, depending on x, such that Σ ∩B r 0 (x) is a C 1,α regular curve (see Theorem 7.16). Problem 1.1 was studied earlier in the particular case p = 2 in [6] for which a full regularity result was proved. We will explain later the differences between our context p = 2 and the Euclidean case p = 2. In the limit p → +∞, Problem 1.1 Γ-converges to the so-called average distance problem which was also widely studied in the literature and for which it is known that minimizers may not be C 1 regular (see [19]). Our result can therefore be considered as making a link between p = 2 and p = +∞, although it actually works for any p > 1.
A constrained variant of the same problem was also studied in [16,17] for p = 2 in dimension 2 and greater, but focusing on different type of questions. In particular, no regularity results were available before with p = 2.
As a matter of fact, even if the present paper is restricted to dimension 2 only, the same problem can be defined in higher dimension, provided that p > N − 1, still with a penalization with the one dimensional Hausdorff measure H 1 (Σ). This instance of the problem in higher dimensions seems to be very original, leading to a free-boundary type problem with a high co-dimensional free boundary set Σ. Due to the low dimension of the "free-boundary" in dimension N > 2, most of the usual competitors are no more valid and some new ideas and new tools have to be used. We believe that some techniques developed in the present paper could be useful to prove a similar result in higher dimensions as well. This will be the purpose of a forthcoming work.
The present paper can therefore be seen as a preliminary step toward the regularity in any dimensions, focusing on the particular case of dimension 2. This approach is pertinent because in dimension 2 only, the "free boundary" Σ is of codimension 1 thus many standard arguments and competitors are available. For instance, one can estimate a p-harmonic function that vanishes on a line in R 2 through a reflection technique, which is no more available for a function in R N with N > 2 which still vanishes on a 1-dimensional line.
However, even in dimension 2, we have to face several technical new difficulties with p = 2 compared to the work for p = 2 in [6] that we shall try to explain now.
Comments about the proof. In the proof of C 1,α regularity, as many other free boundary or free discontinuity problems, one of the main point is to prove a decay estimate on the local p-energy around a flat point. In other words we need to prove that the normalized energy converges to zero sufficiently fast at a point x 0 ∈ Σ, like a power of the radius, and this is where our proof differs from the case p = 2.
In the case p = 2, the decay on the normalized energy is obtained using a so-called monotonicity formula that was inspired by the one of A. Bonnet on the Mumford-Shah functional [2]. This monotonicity formula is also the key tool in the classification of blow-up limits.
For p = 2, an analogous monotonicity formula can still be established for the p-energy (see Section 4). We indeed can prove that r → 1 r α Br(x 0 ) |∇u Σ | p dx + Cr p ′ −α is nondecreasing with a power α depending explicitly on the worst p-Wirtinger constant on the arcs in ∂B r (x 0 ) \ (Σ∪∂Ω), but unfortunately, if x 0 is a flat point, then the resulting power of r in that monotonicity formula is not high enough for our purposes and thus cannot be directly used to prove C 1,α estimates. We stress, for instance, that for p = 2 the monotonicity proof reaches the optimal power α, with equality for the harmonic homogeneous functions with smallest exponent, whereas the monotonicity for p = 2 produces a non-optimal exponent which is not the right power associated to the optimal homogeneous p-harmonic function. Consequently, we also miss a great tool which prevents us to establish the classification of blow-up limits. As the p-monotonicity is not strong enough to get C 1,α regularity we therefore use another strategy, arguing by contradiction and compactness: as mentioned earlier, we know that Br(x 0 ) |∇u Σ | p dx behaves like Cr 2 at point x 0 lying on a line, for a p-harmonic function vanishing on that line (by reflection), thus by compactness it still has a similar behavior when Σ locally stays ε-close to a line. Actually, as the compliance is a min-max type problem, the true quantity to control is not exactly Br(x 0 ) |∇u Σ | p dx, but rather this other variant, as already defined and denoted by ω Σ (x 0 , r) in [6], It can be shown that the quantity ω Σ (x 0 , r) controls, in many circumstances, the square of the flatness, leading to some C 1,α estimates when ω Σ (x 0 , r) decays fast enough.
In [6] the decay of the above quantity was still obtained by use of the monotonicity formula, applied to the function u Σ ′ , where Σ ′ is the maximizer in the definition of ω Σ (x 0 , r).
As a consequence of our compactness argument, which provides a decay only for a closed connected set Σ ′ staying τ -close to a line, we need to introduce and work with the following slightly more complicated quantity where β Σ (x 0 , r) is the flatness defined by (the infimum being taken over all affine lines P passing through x 0 ). We indeed obtain a decay of ω τ Σ (x 0 , r) provided that β Σ (x 0 , r) stays under control, which finally leads to the desired C 1,α result, and the same kind of estimate is also used to prove the absence of loops.
The strategy described above works, but curiously enough, only for the range of exponents p > 4/3. In order to get the regularity result for the full range p > 1, we need some little extra decay. This is obtained by using both the p-monotonicity formula, together with the compactness argument. Indeed, using a dyadic sequence of radii and inserting at each step the p-monotonicity estimate together with the compactness argument, we are able to reach the C 1,α regularity of Σ up to p > 1. for each ϕ ∈ C ∞ 0 (U). We recall the following basic result for weak solutions (see [15]).

Theorem 2.2.
Let U be a bounded open set in R 2 and let u ∈ W 1,p (U). The following two assertions are equivalent.
(i) u is minimizing: (ii) the first variation vanishes: Now we introduce the notion of the Bessel capacity (see e.g. [1], [20]) which is crucial in the investigation of the pointwise behavior of Sobolev functions and in describing the appropriate class of negligible sets with respect to the appropriate Lebesgue measure.
Proof. Let us associate every point Since Φ is a 1-Lipschitz map, by the behavior of Cap p -capacity with respect to a Lipschitz map (see e.g. [1, Theorem 5.2.1]), there is a constant C = C(p) > 0 such that Thus, using (2.1), (2.2) and the facts that Cap p -capacity is an invariant under translations and is monotone with respect to inclusions of sets, we recover the desired inequality.

Uniform boundedness of potentials
Let Ω be a bounded open set in R 2 and let p ∈ (1, +∞). If f ∈ L p ′ (Ω) and Σ is a closed subset of Ω, then it is well-known that there is a unique function u Σ that minimizes E p over W 1,p 0 (Ω \ Σ). We can extend the function u Σ by zero outside Ω \ Σ to an element that belongs to W 1,p (R 2 ). We shall use the same notation for this extension as for the function u Σ . We can prove a boundedness result, uniformly with respect to Σ, for the function u Σ . Proposition 2.10. Let f ∈ L p ′ (Ω) and let Σ ⊂ Ω be a closed set. There is a constant Proof. By using the weak version of the p-Poisson equation which defines u Σ (see (1.2)) we get Applying Hölder and Poincaré inequalities to the right hand side of the previous formula, we get that concludes the proof.

Existence
Theorem 2.11 converge strongly in W 1,p 0 (B), as n → +∞, to the solution of the corresponding problem in Ω. It can be shown that the γ p -convergence is equivalent to the convergence in the sense of Mosco of the associated Sobolev spaces (see [4]). Proof. Let (Σ n ) n be a minimizing sequence for Problem 1.1. Then using Blaschke's selection principle, we can find a compact connected set Σ ⊂ Ω such that up to a subsequence, still denoted by the same index, Σ n converges to Σ with respect to the Hausdorff distance, as n → +∞. Then by Theorem 2.11, u Σ n converges to u Σ strongly in W 1,p 0 (Ω) and thanks to the lower semicontinuity of H 1 with respect to the topology generated by the Hausdorff distance, we deduce that Σ is a minimizer of Problem 1.1.

Dual formulation
Proposition 2.14. Let Ω be a bounded open set in R 2 , p > 1, and let f ∈ L p ′ (Ω). Then Problem 1.1 is equivalent to the minimization problem where B := {(σ, Σ) : Σ ∈ K(Ω) and σ ∈ L p ′ (Ω; R 2 ), −div(σ) = f in D ′ (Ω\Σ)} in the sense that the minimum value of the latter is equal to that of the original Problem 1.1, and once (σ, Σ) ∈ B is a minimizer for (P * ), then Σ solves the original Problem 1.1. Moreover, for a given compact connected set Σ ⊂ Ω, the choice σ : Proof. For the proof we refer to [3].

Ahlfors regularity
We recall that a set Σ ⊂ R 2 is said to be Ahlfors regular, if there exist constants c > 0, r 0 > 0 and C > 0 such that for every r ∈ (0, r 0 ) and for every x ∈ Σ the following holds The notion of Ahlfors regularity is a quantitative and scale-invariant version of having Hausdorff dimension one. It is known that Ahlfors regularity of a closed connected set Σ implies uniform rectifiability of Σ, which provides several useful analytical properties of Σ, see for example [9]. Note that for a closed connected set Σ the lower bound in (3.1) is trivial: indeed, for all x ∈ Σ and for all r ∈ (0, diam(Σ)/2) we have: Σ ∩∂B r (x) = ∅, and then In order to prove Ahlfors regularity for such Σ it suffices to show that there is r 0 > 0, independent of x, such that the upper bound in (3.1) holds for all x ∈ Σ and for all r ∈ (0, r 0 ). Remark 3.1. We need to make some assumptions about regularity of a bounded open set Ω in the statement of Problem 1.1 in order to prove that every solution of this problem is Ahlfors regular. We shall say that a bounded open set Ω in R 2 and its boundary ∂ Ω are locally of class C 0 , if for each point x 0 ∈ ∂ Ω there is a ball B r (x 0 ), an interval I 0 ⊂ R and a continuous simple curve γ 0 : I 0 → R 2 such that ∂ Ω ∩B r (x 0 ) = γ 0 (I 0 ). Let us start by establishing the following topological fact which we shall use in the proof of Ahlfors regularity for a solution of Problem 1.1.

Lemma 3.2.
Let Ω be a bounded open set in R 2 whose boundary ∂ Ω is locally of class C 0 (see Remark 3.1). Then there is r Ω > 0 such that for all r ∈ (0, r Ω ) and for all x ∈ Ω, the set ∂B r (x) ∩ Ω is arcwise connected.

Monotonicity formula
Let Ω be a bounded open set in R 2 . We recall that for any closed set Σ ⊂ Ω the potential u Σ is extended by zero outside Ω \Σ to an element of W 1,p (R 2 ).
In the sequel we shall need the following L p version of the Poincaré-Wirtinger constant By a scaling argument it is easy to see that for an arc of circle S ⊂ ∂B r (0) and for every u ∈ W 1,p 0 (S) it holds S |u| p dH 1 where ∇ τ denotes the tangential gradient. It turns out that the value of λ p was computed explicitly in [7], where it was shown that where Γ is the usual Gamma function. (iii) p → λ p is increasing on (1, 2]; (iv) p → λ p is decreasing on (2, +∞).

Remark 4.3. Let us consider the case when
Notice that when (p, ε, γ) = (2, 0, π), the exponent α in (4.5) equal to one, and actually in this case, using Young's inequality to the term ∂Br(x 0 ) u∇u · ν dH 1 (p = 2) instead of Hölder's inequality that we used in the proof below in (4.6), we can improve α so that it becomes strictly greater than one (see [6,Lemma 3.3]). However, the use of Young's inequality when p = 2 to the term ∂Br(x 0 ) u|∇u| p−2 ∇u·ν dH 1 instead of Hölder's inequality, which we used in (4.6) below, does not give useful results. In fact, using the items (i)-(iv) from Remark 4.1, we see that for all triple (p, ε, γ) in ((1, 2) ∪ (2, +∞)) × [0, 1] × [π, 2π], the exponent α in (4.5) is strictly less than one, and, unfortunately, we cannot improve α in this case. Thus, the power of decay for u Σ provided by α in (4.5), in the case when p ∈ (1, 2) ∪ (2, +∞), is not sufficiently large to use the obtained monotonicity for the investigation the topological structure and the regularity of solutions to Problem 1.1 as it was done in [6]. Nevertheless, we shall use the decay for u Σ provided by the monotonicity formula (4.5) in the proofs of the absence of loops and a C 1,β regularity for solutions of Problem 1.1 in the case when p ∈ (1, 2]. Note that, in fact, the use of this decay in the proofs of absence of loops and a C 1,β regularity for solutions of Problem 1.1 is necessary only in the case when p ∈ (1, 4/3], but to simplify the structure of the proofs, we use it in the case when p ∈ (1, 2]. Proof. Every ball and every circle in this proof is centered at x 0 . For convenience we denote u := u Σ and extend u by zero outside Ω \Σ to an element that belongs to W 1,p (R 2 ). For a.e. r ∈ [2r 0 , r 1 ] we have: Next, thanks to (4.4) and (4.2) we get ∂Br On the other hand, due to (4.4), we have that Σ ∩ ∂B s = ∅ for all s ∈ [r/2, r]. Then, by Corollary 2.9, there is a constant C 0 = C 0 (p) > 0 such that By (4.6)-(4.8) we deduce that where in the latter formula we have used the fact that the function r → G(r) is absolutely continuous on every compact subinterval of (0, +∞) and for a.e. r > 0 its derivative equal to r → ∂Br |∇u| p dH 1 . Then, applying Young's inequality to the term and we deduce that This gives that This concludes the proof.
The following corollary immediately follows from Lemma 4.2 and Remark 4.1.

Decay for the potential u Σ
In this section, we establish the desired decay for the potential u Σ when Σ is flat.

Corollary 5.3.
Let Ω be a bounded open set in R 2 and p > 1. Let f ∈ L p ′ (Ω), Σ and Σ ′ be closed arcwise connected subsets of Ω, and let Then for any r ∈ [r 0 , r 1 /2] we have: for some constant C > 0 depending only on p and f p ′ .
The proof of Corollary 5.3 follows directly from Lemma 5.1 and Lemma 5.2, so we omit it.
and extend them on B 1 using the Schwarz reflection. We show that each of the obtained functions is a weak solution of the corresponding p-Laplace equation in B 1 . Thus we defineũ We claim thatũ and u are weak solutions in B 1 . Indeed, for an arbitrary test function ϕ ∈ C ∞ 0 (B 1 ) we have is a weak solution in int(B + ) and since ψ ∈ W 1,p 0 (int(B + )), using (5.7), we deduce that As ϕ was arbitrarily chosen, we deduce thatũ is a weak solution in B 1 . The proof of the fact that u is a weak solution in B 1 is similar. Thus by [12,Theorem 2.2] there is C = C(p) > 0 such that ess sup Therefore, ess sup that concludes the proof.
Lemma 5.7. Let α and ̺ be as in Corollary 5.6. For any γ ∈ (0, α) there exists ̺ 0 ∈ (0, ̺), depending only on α and γ, such that the following assertion holds. For any closed Proof. For the sake of contradiction, suppose that there exist sequences (ε n ) n , (Σ n ) n and (u n ) n such that ε n ↓ 0 as n → +∞; Σ n is closed, Thus for any n we can define Clearly v n = 0 p-q.e. on Σ n ∩B 1 and By (5.10) and by the fact that (Σ n ∩ B 1 ) ∪ ∂B 1 is connected, there is a constant C 0 > 0 (independent of n) such that for any n large enough we have Then, using the above estimate together with Proposition 2.7 and with (5.13), we conclude that the sequence (v n ) n is bounded in W 1,p (B 1 ). Hence, up to a subsequence still denoted by the same index, we have [4]) that the sequence of Sobolev spaces W 1,p 0 (B 1 \ Σ n ) converges in the sense of Mosco to . Note that by (5.14), v n ψ ⇀ vψ in W 1,p (R 2 ) and using the definition of limit in the sense of Mosco, we deduce that vψ ∈ W 1,p Notice that, in contrary to the linear case, it is not so clear how to pass to the limit in the weak formulation using only the weak convergence of ∇v n to ∇v in L p (B 1 ). But one can argue exactly as in the proof of [4, Proposition 3.7] to get that |∇v n | p−2 ∇v n weakly converges to |∇v| p−2 ∇v in L p ′ (B 1 ), and this is enough to pass to the limit in the weak formulation. We refer to [4] for further details.
We now want to prove the strong convergence of ∇v n to ∇v in L p (B 1/2 ). Since for all n we have that B 1 |∇v n | p dx = 1, we may assume that the sequence |∇v n | p dx of probability measures over B 1 weakly* converges (in the duality with C 0 (B 1 )) to some finite Borel measure µ over B 1 . Then we select some t 0 ∈ (1/2, 3/4) such that µ(∂B t 0 ) = 0. Such t 0 exists, since otherwise µ(∂B t ) > 0 for all t ∈ (1/2, 3/4) and therefore we can find a positive integer number j and an uncountable set of indices A ⊂ (1/2, 3/4) such that for all t ∈ A we have that µ(∂B t ) > 1/j that leads to a contradiction with the fact that µ(B 1 ) < +∞.
From the weak convergence of ∇v n in L p we only need to prove that ∇v n L p (Bt 0 ;R 2 ) tends to ∇v L p (Bt 0 ;R 2 ) . We already have, still by weak convergence, thus it remains to prove the reverse inequality, with a limsup. For this purpose we will use the minimality of v n .
Let χ be smooth cut-off function equal to 1 on B t 0 and zero outside of B 3/4 , and consider the function χv ∈ W 1,p 0 (B 1 \([−1, 1] × {0})). By the definition of convergence in the sense of Mosco, it follows that there is a sequence (ṽ n ) ⊂ W 1,p 0 (B 1 \ Σ n ) converging to χv strongly in W 1,p (B 1 ). Now, fix an arbitrary δ ∈ (0, t 0 −1/2), and let η δ ∈ C ∞ c (B t 0 ) be smooth cut-off function satisfying Then we define In particular, w n = 0 p-q.e. on Σ n and w n = v n outside B t 0 . By the minimality of v n (see Theorem 2.2) we infer Now, recalling that for any ε > 0 there is a constant c ε > 0 such that for all nonnegative real numbers a, b, computing ∇w n and using (5.16) we obtain the following chain of estimates Now notice that since |∇v n | p dx weakly* converges to µ (in the duality with C 0 (B 1 )) and since µ(∂B t 0 ) = 0, we obtain that Bt 0 |∇v n | p dx tends to µ(B t 0 ) as n → +∞ (it is easy to see by taking sequences (g m ) m , (h m ) m ⊂ C 0 (B 1 ) such that g m ↓ 1 Bt 0 , h m ↑ 1 Bt 0 and by using the definition of the weak* convergence of measures). Passing to the limsup, from the strong convergence in L p (B t 0 ) of both v n andṽ n to v we get Letting now δ → 0+ and using the fact that µ(∂B t 0 ) = 0 we get lim sup and we finally conclude by letting ε → 0+ to get lim sup which proves the strong convergence of ∇v n to ∇v in L p (B t 0 ). Using (5.11) and (5.12) and passing to the limit we therefore arrive at On the other hand by Corollary 5.6 applied with u = v and by (5.13) and (5.14) we get which leads to a contradiction with (5.17), since ̺ ∈ (0, 1/2) and γ ∈ (0, α), concluding the proof.
Lemma 5.8. Let α, γ, ̺ and ̺ 0 be as in Lemma 5.7. Let Σ be a closed set in R 2 such that Proof. For convenience we denote S : that concludes the proof. Lemma 5.9. Let α, γ, ̺ and ̺ 0 be as in Lemma 5.7. Let Σ be a closed set in R 2 and let for all r ∈ [r 0 , r 1 ] there is an affine line P , depending on r, passing through Then for any weak solution u of the p-Laplace equation in B r 1 (x 0 )\ Σ such that u = 0 p-q.e. on Σ ∩B r 1 (x 0 ) the following holds Proof. Every ball in this proof is centered at x 0 . Let us fix the nonnegative integer l such that ̺ l+1 r 1 < r 0 ≤ ̺ l r 1 . Then for every r ∈ [r 0 , Then, by iterating the use of Lemma 5.8, we deduce that Thus, that concludes the proof.
We will need the following lemma (see [8,Lemma 2.2]), wich we shall use to establish control on the difference between the potential u Σ and its Dirichlet replacement on a ball with crack. For convenience we use the notation a(x, ξ) := |ξ| p−2 ξ, x, ξ ∈ R 2 .
Proof. Every ball in this proof is centered at x 0 . For convenience let us define z := u Σ −w.
Since z ∈ W 1,p (B r 1 (x 0 )), z = 0 p-q.e. on Σ ∩B r 1 (x 0 ), by Corollary 2.9, that can be applied because Σ is arcwise connected and because of (5.21), we have where C 1 depends only on p. We consider the next two cases. Case 1: 2 < p < +∞. Using (5.19) and the fact that z is a test function for (5.22) and (5.23), we get where c 0 = c 0 (p) > 0. Applying Hölder inequality to the right-hand side of the latter formula, we obtain where the latter estimate comes by using (5.26). Therefore, Next, by using Hölder's inequality and then (5.26), we obtain where the latter estimate comes from the fact that w minimizes the energy Br 1 |∇v| p dx among all v satisfying v − u Σ ∈ W 1,p 0 (B r 1 \Σ) and u Σ is a competitor for w. Therefore, that yields (5.25).
Proof. In this proof every ball is centered at x 0 . Let w be the Dirichlet replacement of u Σ in B r 1 , i.e. w is a weak solution of the p-Laplace equation in B r 1 \ Σ with Dirichlet condition w = u Σ on ∂B r 1 ∪ (Σ ∩B r 1 ). This function is found by minimizing the energy The latter inequality comes from the fact that w minimizes the energy in B r 1 and u Σ is a competitor. By convexity (1 < p < +∞) for any r ∈ [r 0 , r 1 ] we deduce that Next, by using Lemma 5.11 for an appropriate p ∈ (1, +∞), we estimate the second term on the right-hand side of the latter inequality, that finally yields (5.27) and (5.28).

Absence of loops
Then every solution Σ of the penalized Problem 1.1 contains no closed curves (homeomorphic images of S 1 ), hence R 2 \ Σ is connected.
The next two geometric lemmas will be used in the proof of the theorem. (ii) Σ \D n are connected for all n; (iii) diam D n ց 0 as n → +∞; (iv) D n are connected for all n.
Then, by Corollary 4.4, we get where C 0 = C 0 (p, f p ′ , ε, r 0 ). By (6.12) and (6.13), we get and q > p ′ . Thus, choosing α ∈ (0, 1) very close to one and choosing ε very close to zero, we can find γ ∈ (0, α) such that Then we can find a constant s > 0 such that where the latter estimate implies that (1 − s)(1 + γ) + 2s(1 − 4ε)/(π + 4ε) > 1. Thus, F λ,p (Σ n ) − F λ,p (Σ) = Cg(r n ) − λr n , where g(r n ) = o(r n ) as n → +∞, that leads to a contradiction with the fact that Σ is a minimizer and hence prove that Σ contains no closed curves. In order to prove the last assertion in Theorem 6.1, we use theorem II.5 of [14, § 61], stating that if D ⊂ R 2 is a bounded connected set with locally connected boundary, then there is a simple closed curve S ⊂ ∂D. If R 2 \ Σ were disconnected, then there would exist a bounded connected component D of R 2 \Σ such that ∂D ⊂ Σ, and hence Σ would contain a simple closed curve, contrary to what we proved before.

Proof of a C 1,α regularity
In this section, we shall prove that every solution of Problem 1.1 is locally a.e. C 1,α regular. For the proof we shall partially use the strategy given for the linear case in [6]. However the big difference between the proof given in [6] and the one we present is that in [6] a decay of the energy, provided that Σ stays flat, was obtained using the "monotonicity formula" (for the definition see [6,Lemma 3.3]), but as we have seen in Remark 4.3 for all p ∈ (1, 2) ∪ (2, +∞) the power of decay for the potential u Σ around a flat point provided by the "monotonicity formula" (4.5) is small and we cannot directly use it to study the regularity of a solution Σ of Problem 1.1. Whereas in our proof we introduce a new definition for the energy w τ Σ , that takes into account the flatness, and we use a decay for the potential u Σ obtained in Section 5, that, due to the lack of a sufficient power of decay in the "monotonicity formula" (4.5) at p = 2, makes it possible to establish the desired decay for the energy w τ Σ provided that Σ stays flat and then using a control on the flatness when the energy w τ Σ is small, leads to the so-called ε-regularity theorem. Note that in the proof of the regularity we do not use the decay for the potential provided by "monotonicity formula" (4.5) in the case when 2 < p < +∞ and we need to use it together with the decay established in Section 5 only in the case when p ∈ (1, 4/3], however, so as to simplify the structure of the proof, we use it in the case when p ∈ (1, 2].
It is clear that the factor λ in the statement of Problem 1.1 affects the shape of an optimal set minimizing functional F λ,p , however, it does not affect the regularity of this optimal set. Thus, in this section, we shall assume that λ = 1 for convenience.
The proof of this proposition is exactly the same as the one of Lemma 5.12. Let us now introduce the following definition of the energy. Definition 7.2. Let Σ ∈ K(Ω) and let τ ∈ (0, 1). For any x 0 ∈ R 2 and any r > 0, define Remark 7.3. If β Σ (x 0 , r 1 ) < ε with ε ∈ (0, τ ) and τ ∈ (0, 1), then for any r ∈ [ εr 1 τ , r 1 ] there is a solution for the problem (7.5). Indeed, using (7.1), we deduce that τ , r 1 ] and hence Σ is an admissible set in (7.5). Then by Proposition 2.10, w τ Σ (x 0 , r) < +∞. If Σ k be a maximizing sequence for w τ Σ (x 0 , r), then up to a subsequence Σ k → Σ 0 for the Hausdorff distance and clearly Σ 0 ∆ Σ ⊂ B r (x 0 ). Let P k be the line that realizes the infimum in the definition of β Σ k (x 0 , r). Then, possibly passing to a subsequence still denoted by k, one has that (7.6) for some affine line P 0 passing through x 0 . Since for any k : using (7.6), we deduce that β Σ 0 (x 0 , r) ≤ τ . So, Σ 0 is an admissible set for (7.5). Futhermore, by Theorem 2.11, ∇u Σ k converges strongly to ∇u Σ 0 in L p (R 2 ) and Σ 0 is therefore a maximizer.
In order to establish a decay for w τ Σ , we need the following geometrical result.
Proposition 7.4. Let Σ ∈ K(Ω), x ∈ Ω and τ ∈ (0, 1), and let β Σ (x, r 1 ) ≤ ε for some ε ∈ (0, τ ). Then for all r ∈ [εr 1 /τ, r 1 ], for all Σ ′ ∈ K(Ω) such that Σ ′ ∆ Σ ⊂ B r (x) and β Σ ′ (x, r) ≤ τ , one has: Proof. Every ball in this proof is centered at x. By using (7.1) we deduce that β Σ (x, t) ≤ τ for all t ∈ [εr 1 /τ, r 1 ]. (7.9) Let P 0 be the line that realizes the infimum in the definition of β Σ (x, r). Then by (7.9), Let P 1 be the line realizing the infimum in the definition of β Σ ′ (x, r) and let P 2 be the line realizing the infimum in the definition of β Σ (x, r 1 ). Then we observe the following: (7.11) where the latter inequality is due to the facts that Σ ′ ∆ Σ ⊂ B r and β Σ (x, r 1 ) ≤ ε. In addition, and given that Σ ′ ∈ K(Ω) and Σ ′ ∆ Σ ⊂ B r , we can estimate d H (P 0 ∩ B r , P 1 ∩ B r ) as follows Thus By using (7.10), (7.12) and the fact that β Σ ′ (x, r) ≤ τ we deduce that This together with (7.11) gives the following Thus we proved (i). Now let s ∈ [r, r 1 ] and let P s be the line realizing the infimum in the definition of β Σ (x, s). As in the proof of (i) we get Then by (7.13) and (7.9) we deduce that thus concluding the proof.
Proof. Every ball in this proof is centered at x 0 . By Corollary 5.3, (7.17) where C = C(p, f q ) > 0. Now recall that by Proposition 2.10 the constant K(u Σ , u Σ ) in (7.4) is uniformly bounded with respect to Σ ∈ K(Ω). Then, using (7.17) and Proposition 7.1, we deduce the following chain of estimates (with the constant C changing from estimate to estimate) (by Corollary 7.5 (i) and by definition of w τ Σ ), where the last one together with the fact that r < r 1 leads to (7.16).
(ii) Assume that the estimate is valid with ν and C such that ( (iii-2) Σ ∩B r (x 0 ) is an arcwise connected.
Remark 7.9. Following [6], if the situation of item (iii-1) occurs we say that the two points lie "on both sides".
Proof. Every ball in this proof is centered at x 0 .
Step 1. We prove (i). From (7.1) and (7.18) we deduce that for all r ∈ [τ r 1 /4, r 1 /12]. Let us fix an arbitrary r ∈ [τ r 1 /4, r 1 /12]. Let ξ 1 and ξ 2 be the two points of ∂B r ∩ P 0 , where P 0 is a line that realizes the infimum in the definition of β Σ (x 0 , r) and let W be the set defined by We consider the set Clearly, Σ ′ ∆ Σ ⊂ B r and Σ ′ is arcwise connected and compact. In addition, from (7.23) it follows that β Σ ′ (x 0 , r) < τ.
. Arguing in the same way as in the proof of the fact that E 1 ⊂ (0, τ r 1 /100) in Step 2, we deduce that On the other hand, as in Step 1 we have that which leads to a contradiction with the fact that H 1 (Σ ∩B r ) ≥ r ≥ τ r 1 /4, therefore (iii-1) holds. Next assume that Σ ∩B r is not connected. Then, from [6, Lemma 5.13], it follows that Σ \B r is an arcwise connected. Thus, taking the set Σ ′ := Σ \B r as a competitor, we get H 1 (Σ ∩ B r ) ≤ τ r 1 /150, that leads to a contradiction with the fact that H 1 (Σ ∩B r ) ≥ τ r 1 /4. Thus (iii-2) holds. Since Σ ∩∂B r = {z 1 , z 2 }, where z 1 , z 2 lie on "on both sides", the set Σ \B r ∪[z 1 , z 2 ] is a competitor for Σ. On the other hand since w τ Σ (x 0 , r 1 ) < 1, then w τ Σ (x 0 , r 1 ) ≤ (w τ Σ (x 0 , r 1 )) 1/p , which leads to the estimate (7.22). So the proof of (iii) is concluded.

Remark 7.10.
If Ω ⊂ R 2 is a C 1 domain and Σ ⊂ Ω is a minimizer for Problem 1.1, then, using Ahlfors regularity of Σ, we can show that the assertions (i), (ii), (iii-1) and (iii-2) holds for every x 0 ∈ Σ. For the proof in the linear case (when p = 2) we refer to [6], the proof for the nonlinear case is similar.

Control of the flatness
We recall the following standard height estimate (see [6,Lemma 5.14]), which we shall use so as to establish a control on β Σ via w τ Σ .
Next theorem says that Theorem 1.2 still holds at the boundary if the domain is convex and regular. We recall that u Σ is extended by zero outside Ω \ Σ to an element that belongs to W 1,p (R 2 ). Theorem 7.16. Let Ω ⊂ R 2 be a convex domain of class C 1 , p ∈ (1, +∞), f ∈ L q (Ω) with q > 2 if 2 < p < +∞ and q > p ′ if 1 < p ≤ 2, and let Σ ⊂ Ω be a minimizer for Problem 1.1. Then there is a constant α ∈ (0, 1) such that for H 1 -a.e. point x ∈ Σ one can find a constant r 0 > 0, depending on x, such that Σ ∩B r 0 (x) is a C 1,α regular curve.
Proof. Due to convexity of Ω we can relax the class of competitors for Problem 1.1, namely we can change the class {Σ ⊂ Ω, closed and connected} by the class {Σ ⊂ R 2 , closed and connected}. The projection P Ω onto Ω is 1-Lipschitz thus minimizing F 1,p on the second class would lead to the same minimizers, since the projection of any competitor has a lower value of F 1,p . So, in the case of convex domains we can consider Σ as a subset of R 2 with competitors in R 2 . Then, thanks to Remark 7.10, all proofs in Section 7 "work" at the boundary straight away.

Acknowledgments
This work was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR).

A Auxiliary results
In the next lemma we prove the integration by parts formula for a weak solution of the p-Poisson equation. Then for every x 0 ∈ R 2 and a.e. r > 0 we have where ν stands for the normal to ∂B r (x 0 ).