Mass concentration in rescaled first order integral functionals

We consider first order local minimization problems of the form $\min \int_{\mathbb{R}^N}f(u,\nabla u)$ under a mass constraint $\int_{\mathbb{R}^N}u=m$. We prove that the minimal energy function $H(m)$ is always concave, and that relevant rescalings of the energy, depending on a small parameter $\varepsilon$, $\Gamma$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_i\delta_{x_i}$ as $\sum_i H(m_i)$. We also consider Lagrangians depending on $\varepsilon$, as well as space-inhomogeneous Lagrangians and $H$-masses. Our result holds under mild assumptions on $f$, and covers in particular $\alpha$-masses in any dimension $N\geq 2$ for exponents $\alpha$ above a critical threshold, and all concave $H$-masses in dimension $N=1$. Our result yields in particular the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.

(1.1) The minimization of this energy under a mass constraint gives rise to the notion of minimal cost function, valued in [0, +∞] and defined by As a preliminary result, which applies to (1.1) and deserves interest on its own, we will establish the following: Theorem 1.1.-Let f : R×R N → [0, +∞] be Borel measurable such that f (0, 0) = 0.The function defined for every m ∈ R by vanishes at 0 and it is either identically +∞ on (0, +∞), or it is everywhere finite, continuous, concave and non-decreasing on [0, +∞).The symmetric statement on (−∞, 0] holds as well.
J.É.P. -M., 2024, tome 11 The proof is very simple and works with no further assumptions on f .Our main purpose is to prove that if (f ε ) ε>0 is a family of functions which converges pointwise to f as ε → 0 and satisfies some conditions, then the rescaled energy functionals E ε , defined for each ε > 0 on M (R N ) by (1.4) Γ-converge as ε → 0, for the narrow or weak convergence of measures, to the H f -mass, defined on M (R N ) by (see Definition 2.2): where µ = µ a + µ d is the decomposition of µ into its atomic part µ a and its diffuse part µ d , µ d = µ d + − µ d − is the Jordan decomposition of µ d into positive and negative parts, and H ′ f (x, 0 ± ) is the recession at 0, that is Notice that the exponents over ε in the definition of E ε are tuned so that if B r (x 0 ) ⊆ R N and u ε is a mass-preserving rescaling of v ε given by u ε (x) =: ˆBr/ε f ε (x 0 + εy, v ε (y), ∇v ε (y)) dy, so that the energy contribution of a mass m ⩾ 0 contained in a ball B r (x 0 ) should be of the order of H f (x 0 , m).This explains why M H f is expected to be the Γ-limit of E ε .Nevertheless, it is not true in general (see Section 1.3 below), and we will need a couple of assumptions on f and f ε detailed in the next section.This kind of singular limit of integral functionals is reminiscent of several variational models with physical relevance which have been the object of intensive mathematical analysis, such as Cahn-Hilliard fluids with concentration on droplets [BDS96] (which we recover in Section 5.5) or on singular interfaces [MM77], toy models for micromagnetism and liquid crystals like Aviles-Giga [AG99] and Landau-de Gennes [BPP12], or Ginzburg-Landau theory of superconductivity [BBH17].
1.2.Assumptions and main result.-Our first two assumptions are rather standard and guarantee the sequential lower semicontinuity of the functionals E x f , (H 1 ) f : R N × R × R N → [0, +∞] is lower semicontinuous, (H 2 ) f (x, u, •) is convex for every x ∈ R N , u ∈ R. In order for vanishing parts to have no energetic contribution, we will impose (H 3 ) f (x, 0, 0) = 0 for every x ∈ R N .
We also need continuity in the spatial variable, (H 4 ) f (•, u, ξ) is continuous for every u ∈ R, ξ ∈ R N .
Next, we need a compactness assumption which ensures relative compactness in the weak topology of W 1,p loc (R N ) for sequences of bounded energy E x f and bounded mass; it will also be needed in obtaining lower bounds for the energy (see Proposition 3.8): (H 5 ) there exist α, β ∈ (0, +∞), p ∈ (1, +∞) such that for all (x, u, ξ) ∈ R N ×R×R N , f (x, u, ξ) ⩾ α|ξ| p − βu.
We also impose a comparison condition on the slopes of f (x, •, ξ) and H f (x, •) at the origin, which will be needed in order to show that the Γ-lim inf is bounded from below by the H f -mass, and which rules out some non-trivial scale invariant Lagrangians for which the expected Γ-convergence result fails (see Section 1.3): We give a general assumption (simple in dimension one but quite technical in dimension N ⩾ 2) depending only on the Lagrangian f so as to guarantee such a condition (see (S) and Corollary 2.7 in Section 2.3).Since our aim is not to care much about the dependence on x, we shall impose a spatial quasi-homogeneity condition: (H 7 ) there exists C < +∞ such that for every x, y ∈ R N , u ∈ R, ξ ∈ R N , f (y, u, ξ) ⩽ C(f (x, u, ξ) + u).
Our main result is the following: Theorem 1.2.-If (f ε ) ε>0 satisfies (H 8 ) with each f ε satisfying (H 1 )-(H 5 ) and the limit f satisfying (H 6 )-(H 7 ), then M H f is the Γ-limit as ε → 0 of the functionals E ε , defined in (1.4), for both the weak convergence and the narrow convergence of measures.
In particular, as a Γ-limit, the functional M H f must be lower semicontinuous for the weak convergence of measures (and so for the narrow convergence as well).This implies that H f is lower semicontinuous on R N × R (see Proposition 2.4).
We point out that for the Γ-lim sup, we need weaker assumptions on f ε and f (see Proposition 4.2), which will be useful for some applications (see Section 5.5).
J.É.P. -M., 2024, tome 11 We will allow ourselves slight abuses of notation.First of all, we will sometimes consider Lagrangians defined on R × R N which do not depend on x and still refer to hypotheses (H 1 )-(H 8 ) ; we will use the notation E f instead of E x f in (1.1) and consider H f as a function of m only in (1.2).Besides, we will often identify functions u ∈ L 1 loc with the measures µ = uL N , so as to concisely write Finally, we will also consider Lagrangians defined only for u ∈ R + , which may be thought as defined for u ∈ R, set to +∞ when u is negative. (1)The resulting minimal cost function H f and its associated H f -mass may be thought as defined on R + and M + (R N ) respectively, as they will be infinite on the respective complements.
1.3.Remarks and applications.-We start with two situations where the expected Γ-convergence fails and which justify the importance of (H 3 ) and (H 6 ), then we provide examples and applications of our result, as a short summary of Section 5, where full details are provided.We restrict our attention to positive measures and Lagrangians defined for u ⩾ 0.
Vanishing parts do not contribute to energy.-By assumption (H 3 ) no energy is given to any set where a function u vanishes.It is a necessary condition for M H f to be lower semicontinuous (a necessary condition to be a Γ-limit) and not identically +∞.Indeed if M H f is lower semicontinuous and finite for some measure , 0, 0) this can only happen if f (•, 0, 0) ≡ 0. This justifies imposing (H 3 ).

Scale invariant
Lagrangians.-In Section 5.1, we will see that in the particular case f ε (x, u, ξ) = u −p(1−1/p ⋆ ) |ξ| p for p ∈ (1, N ) and p ⋆ = pN /(N − p), our Γ-convergence result does not hold as a consequence of the fact that E ε does not depend on ε.Note that in this case, the slope assumption (H 6 ) does not hold, and we also provide a simple variant of such energies which satisfies all our assumptions except (H 6 ) where the Γ-convergence towards M H f also fails, thus justifying the need for such a slope condition.
Concave H-masses in dimension one.-Consider the energy given for µ = u L N by In dimension N = 1, it is shown in [Wir19] that for any concave continuous function H with H(0) = 0, there exists a suitable c ⩾ 0 such that H f = H.As explained in Section 5.2, Theorem 1.2 implies that the rescaled energies (1) Notice that if any of our assumptions is satisfied for a Lagrangian defined for u ∈ R + , then it holds also for the Lagrangian extended to R in this way.
Γ-converge to M H , leading to an elliptic approximation of any concave H-mass in dimension one.In dimension N ⩾ 2, we will show that H f must be concave on [0, +∞), and strictly concave after the possible initial interval where it is linear (see Proposition 2.8) ; however, we have no solution to the inverse problem, consisting in characterizing the class of attainable minimal cost functions H = H f for Lagrangians f satisfying our assumptions.
Homogeneous H-masses in any dimension.-We consider the functional given for µ = u L N by Then, the rescaled energies varies in its range and N ⩾ 1.More details are given in Section 5.3.
Cahn-Hilliard approximations of droplets models.-Following the works of [BDS96,Dub98], we consider the functionals where W (t) ∼ t→+∞ t s for some exponent s ∈ (−2, 1).As shown in Section 5.5, we way rewrite these functionals to fit our general framework, and recover known Γ-convergence results, under slightly more general assumptions, as stated in Theorem 5.1.The Γ-limit is a non-trivial multiple of the α-mass with α = 1−s/2+s/N 1−s/2+1/N .Elliptic approximations of branched transport.-The energy of branched transport (see [BCM09] for an account of the theory), in its Eulerian formulation, is an H-mass defined this time on vector measures w whose divergence is also a measure, (1.9) where w = θξ •H 1 Σ+w ⊥ is the decomposition of w into its 1-rectifiable and 1-diffuse parts (see Section 5.4 for more details).An elliptic approximation of Modica-Mortola type has been introduced in [OS11] for H(m) = m α , α ∈ (0, 1), and their Γ-convergence result in dimension d = 2 has been extended to any dimension in [Mon15] by a slicing method which relates the energy of w to the energy of its slicings.The same slicing method, together with Theorem 1.2, would allow to prove the Γ-convergence of the functionals (1.10) , thus covering a wide range of concave H-masses over vector measures with divergence.
1.4.Structure of the paper.-In Section 2, we prove the concavity of the minimal cost function H f with respect to the mass variable m in full generality (Theorem 1.1), we establish useful properties of general H-masses, and we identify the slope at the origin of H f in terms of f under our assumption (Proposition 2.5 and Proposition 2.6).In Section 3, we apply a concentration-compactness principle to provide a profile decomposition theorem for sequences of positive measures (Theorem 3.2), which is used to obtain our main lower bound for the energy E f (Proposition 3.10) and also yields an existence criterion for profiles with minimal energy under a mass constraint (Proposition 3.12).Section 4 is dedicated to proving lower and upper bounds on the rescaled energies E ε (Proposition 4.1 and Proposition 4.2) that imply in particular our main Γ-convergence result (Theorem 1.2).Last of all, in Section 5, we provide counterexamples and several examples of energy functionals that fall into our framework, as summarized in the previous section.

Minimal cost function and H-mass
In this section, we study the properties of general H-masses, of costs H f associated with general Lagrangians f , and we relate the slope of H f at m = 0 to that of f at (u, ξ) = (0, 0) in the variable u, under particular conditions.
2.1.Concavity and lower semicontinuity of the minimal cost function.-Before proving Theorem 1.1, let us note that it covers the case where we have a constraint (u, ∇u) ∈ A, where A ⊆ R × R N is Borel measurable, by considering Lagrangians f taking infinite values.
Proof of Theorem 1.1.-Let us first assume that f (u, ξ) = +∞ when u < 0, so as to restrict ourselves to non-negative functions.We first prove that H f is concave on (0, +∞).Let m > 0 and u ∈ W 1,1 loc ∩ L 1 (R N , R + ) such that ´RN u = m.We pick a non-zero vector v ∈ R N and for every t ∈ R, we set u t (•) = u(• + tv) and Moreover, it is standard that we have similar identities for u ∨ u t , and we obtain Let M : t → ´RN u ∧ u t .In view of (2.1), (2.2), and by definition of H, we have proved By continuity of translations in L 1 and since the map that is, H f is midpoint concave on (0, +∞).Since H f is also bounded below (by 0), we can deduce that H f is concave (0, +∞) (see [RV73,§72]), and since H f ⩾ 0, either H f is identically +∞ on (0, +∞), or it is finite everywhere, continuous, concave and non-decreasing on (0, +∞).We now justify that if H f (m) < +∞ for some m > 0 and f (0, 0) = 0, then Since M is continuous with M (0) = ´RN u > 0 and lim t→+∞ M (t) = 0 as seen above, we have that t * ∈ (0, +∞], lim t→t * M (t) = 0 and M (t) does not vanish identically near t * .Moreover, if t * = +∞, since u t → 0 locally in measure, by dominated convergence, ⩽ 2f (0, 0)|{u = 0}| = 0.
Finally, we treat the general case (without assuming that f (u, ξ) = +∞ if u < 0) For this, notice that for every u ∈ W 1,1 loc ∩L 1 (R N ) such that ´RN u = m ⩾ 0, denoting by u + the positive part of u and m -Let us remark two things about the proof.First, we actually proved the concavity and monotonicity of on (0, +∞) for f Borel, without assuming f (0, 0) = 0.
Second, the end of the proof shows that under this extra assumption, minimizing among signed profiles or non-negative (resp.non-positive) profiles is equivalent when m ⩾ 0 (resp m ⩽ 0).

Definition and relaxation of the H-mass
Definition 2.2.-Let H : R N × R → [0, +∞] be a Borel measurable function having left/right slopes at the origin defined for each x ∈ R N by (2.4) For every finite signed Borel measure µ ∈ M (R N ), we set , where µ = µ a + µ d is the decomposition of u into its atomic part µ a and its diffuse (or non-atomic) part µ d , µ d = µ d + − µ d − is the Jordan decomposition of µ d into positive and negative parts.
The H-mass of µ is then defined as the total variation of H(µ), that is: The previous definitions and notations extend in the obvious way to the case of functions H : R → [0, +∞] with no space variable x, interpreted as functions independent from x.
M H is a natural spatially non-homogeneous extension (depending on the position x) of the H-mass of k-dimensional flat currents (2) from Geometric Measure Theory, introduced by [Fle66] (see also the more recent works [DPH03,CDRMS17]).
From another work from the same authors [BB93, Th. 3.2], we know that under some further assumptions on H, M H is the relaxation for the weak topology of the functional We need a slightly different result, (4) namely that for any function H : R N × R → [0, +∞] satisfying all the assumptions of Proposition 2.3 except the lower semicontinuity, the relaxation of M H atom for the narrow sequential convergence is M H lsc , where H lsc is the lower semicontinuous envelope of H, which can be expressed as It is worth noticing that if H(•, 0) ≡ 0 and H is mass-concave, then these properties hold also for H lsc .
Proposition 2.4.-Let H : R N × R → [0, +∞] be a function which is mass-concave and such that H(•, 0) ≡ 0.Then, the sequentially lower semicontinuous envelope of M H atom in the narrow topology of M (R N ) is given by M H lsc , namely we have: We point out that for a general H, for M H to be sequentially lower semicontinuous (for the narrow topology) it is necessary that H is lsc on R N × (0, +∞).However, neither the subadditivity of H in m nor its lower semicontinuity on R N ×R + are necessary.Indeed, M H is sequentially lower semicontinuous if for instance H(x, m) = +∞ when x ̸ = 0, m > 0, H(x, 0) = 0 when x ̸ = 0 and H(0, •) is any lower semicontinuous (2) In the case k = 0, since signed measures are merely 0-currents with finite mass.
(3) In the notation of this paper, we take µ = 0 and f (x, s) = |s| 2 ; we have φ f (x, 0) = 0 and H is assumed to be lower semicontinuous and the authors make a further coercivity assumption (assumption (3.5) in the paper) that we want to avoid.
We first prove that For this, we let µ = k i=1 m i δ xi be a finitely atomic positive measure and we let Then (µ n ) n∈N converges narrowly to µ and, by lower semicontinuity, so that F (µ) ⩽ M H lsc atom (µ) as wanted.We now prove that F ⩽ M H lsc .Let µ ∈ M (R N ) and µ = µ a + µ d be the decomposition of µ into its atomic part µ a = k i=1 m i δ xi , with k ∈ N ∪ {+∞} (here, k = 0 if there is no atom), and its diffuse part µ d , and let µ d = µ d + − µ d − be the Jordan decomposition of µ d into positive and negative parts.We then discretize µ d

and we define
where for each i ∈ {1, . . ., Such points exist since Q n i is compact and since by concavity, (2.9) so that H ′ lsc (•, 0 ± ) are lower semicontinuous as suprema of lower semicontinuous functions.
The sequence (µ n ) n∈N converges narrowly to µ.We deduce by lower semicontinuity of F , (2.9) where we have used monotone convergence in the last but one equality.□ 2.3.Slope at the origin of the minimal cost function.
-In this section we provide a technical assumption, that is simple in dimension N = 1, on the Lagrangian f and which implies the comparison condition on the slopes (H 6 ).
We now consider the solution of the ODE because it is nonincreasing and bounded, hence it has finite total variation, and because it is of class C 1 except possibly at r ε := F (0) − F (ε), where it has no jump.As a consequence the radial profile defined by u ε (x) := v ε (|x|) belongs to W 1,1 loc (R N ) and we compute, using the change of variables s = v ε (r) (i.e., r = F (s) − F (ε)) and an integration by parts combined with monotone convergence.
The equality on the last line holds because N exists by existence of the limit in the previous line, and it must be zero (again, because By assumption, we deduce that In dimension N = 1, we need no other assumption than H f < +∞, as stated below. Proposition 2.6.-Let f : R + × R N → [0, +∞] be Borel measurable with N = 1.The minimal cost function H f is either identically infinite on (0, +∞), or it satisfies (2.12) with ρ ≡ 0, i.e., Proof.
-One can assume that there exists u ∈ W 1,1 loc (R, R + ) with 0 < ´R u < +∞ and E(u) < +∞.In particular, up to changing the value of u on a negligible set, u is continuous on R. Let ε ∈ (0, sup R u), set A ε := {x : u(x) = ε} which is non-empty by the intermediate value theorem and integrability of u, and define By continuity and integrability of u, u(a and by dominated convergence, since ∇u = 0 a.e. on {u = 0}, Notice that this limit is necessary zero.Let m > 0. If ε is small enough, then ´R∖[aε,bε] u < m so that we can take We then define From Proposition 2.5 and Proposition 2.6 we obtain the following corollary. Proof.
-If N = 1 we apply Proposition 2.6 to get and if N = 2 taking a function ρ as in (S) and applying Proposition 2.5 yields We show that in dimension N ⩾ 2, the minimal cost function must be strictly concave away from the possible initial interval where it is linear: where H f is defined in (1.3).Then, H f is strictly concave on (m * , +∞).A similar statement holds on R − .
A similar result does not hold in dimension 1 since any continuous concave function H : R + → R + with H(0) = 0 can be written as H = H f with f satisfying all our assumptions (H 1 )-(H 8 ) (see Section 5.2).
We denote by M f m the set of non-negative minimizers of mass m ∈ R + : (2.13) The proof of Proposition 2.8 is based on the following observation: Lemma 2.9.-Let f : R + ×R N → [0, +∞] be Borel measurable and let Proof of Lemma 2.9.-We use the same observations as in the proof of Theorem 1.1.
In particular, we have But we have also so that the inequalities we used, i.e., We also use an elementary Sobolev type inequality: .
Proof of Lemma 2.10.-We prove the lemma when u ∈ C 1 (R × ω); the general case follows by approximation.For every x 1 , y 1 ∈ R, x ′ ∈ ω, we have By averaging in the variable y 1 , we deduce The result follows from Hölder inequality after integrating over ω.
As before, we shall use the notations ∧ and ∨ for the minimum and maximum; we also let (e 1 , . . ., e N ) be the canonical basis of and since the set M m+η/2 is compact in L 1 up to translations in view of Remark 3.13, we deduce that there exists δ 0 > 0 such that > m for all δ ∈ (0, δ 0 ) and u ∈ M f m+η/2 .We now construct by induction a sequence (t n ) n∈N in R + and a sequence where we have set U (x) := ess sup t∈R u(x + te 1 ).
To this aim, we first set u 0 := u and t 0 = 0.Then, if we assume that t n and u n are constructed as before, we first pick an δ 1 n ∈ R + such that which is possible since η ⩽ m, as we argued in the proof of Theorem 1.1.Similarly, we pick a where in the last inequality we have used the induction hypothesis (2.15).We now show that the sequence (u n L N ) n∈N is vanishing which will contradict the compactness of M f m in L 1 up to translations.
J.É.P. -M., 2024, tome 11 For this, we let (x k ) k∈N be a sequence in R N and (u n k ) k∈N be a subsequence of By (H 5 ), we have ∂u/∂x 1 ∈ L p (R N ).Using this fact, estimate (2.15), and Lemma 2.10 with ω a unit cube in R N −1 , we obtain and the conclusion follows since the sequences (x k • e 2 ) k∈N and (t k∈N cannot be both bounded as lim k→∞ t n k = ∞.□

Lower bound for the energy and existence of optimal profiles
Our main tool to localize the energy and obtain a lower bound relies on a profile decomposition for bounded sequences of positive measures, which is reminiscent of the concentration-compactness principle of P.-L.Lions.This differs from classical strategies to localize the energy which are based on suitable cut-offs.Naturally, this concentration-compactness result also provides a criterion for the existence of optimal profiles in (1.2).Nothing can be said beyond existence of a minimizer at this level of generality.Further properties such as uniqueness and radial symmetry would require conditions on the Lagrangian f and not only on the cost function H f .We deal with these questions in a particular case in Section 5.3.

3.1.
Profile decomposition by concentration-compactness. -We prove a profile decomposition theorem for bounded sequences of positive measures over R N , which is essentially equivalent to [Mar14, Th. 1.5] in the Euclidean case.We have added an extra information on mass conservation that will be useful, and provide a selfcontained simple proof.We start with a definition.
Any bounded sequence of positive measures over R N may be decomposed (up to subsequence) into a countable collection of narrowly converging "bubbles" and a vanishing part, accounting for the total mass of the sequence, as stated in the following theorem.
Theorem 3.2.-For every bounded sequence (µ n ) n∈N of positive Borel measures on R N , there exists a subsequence (µ n ) n∈σ(N) , σ ∈ Σ, a non-decreasing sequence of integers (k n ) n∈σ(N) converging to some k ∈ N ∪ {+∞}, a sequence of non-trivial positive Borel measures (µ i ) 0⩽i<k , and for every n ∈ σ(N), a collection of balls (B i n ) 0⩽i<kn centered at points of supp µ n such that, writing for all n ∈ σ(N), (B) bubbles split: Before proving Theorem 3.2, we introduce the "bubbling" function of a sequence of finite signed measures (µ n ) n∈N : Although we will use this function on signed measures, we will start from a sequence of positive measures and use the following characterization of vanishing sequences, which holds only in the case of positive measures: Proof.-Assume that (µ n ) n∈N is vanishing and that (τ −x σ(ℓ) µ σ(ℓ) ) ℓ∈N C ′ 0 − − ⇀ µ for some σ ∈ Σ and some sequence of points (x σ(ℓ) ) ℓ∈N .Then, for every i.e., µ = 0 and thus m((µ ℓ ) ℓ∈N ) = 0. Conversely, if (µ n ) n∈N is not vanishing, then there exists ε > 0, σ ∈ Σ and a sequence of points Up to further extraction, one can assume that n∈N is vanishing, then we take σ = Id and k = 0, so that µ σ(ℓ) = µ ℓ = µ v ℓ , (A) to (D) are empty statements and (E) is satisfied since (µ n ) n∈N is vanishing.Assume on the contrary that (µ n ) n∈N is not vanishing.We shall construct the bubbles by induction and prove their properties in several steps.
J.É.P. -M., 2024, tome 11 Step 1: construction of bubbles centers.-At first step (step 0), since m((µ n ) n∈N ) > 0, there exists σ 0 ∈ Σ and a sequence of points (x 0 n ) n∈σ0(N) , such that We then set µ 0 n := µ n −τ x 0 n µ 0 and we continue by induction, starting from the sequence (µ 0 n ) n∈σ0(N) .More precisely, assume that for a fixed step Either the induction stops at some step k − 1 ∈ N for which m((µ k−1 n ) n∈σ k−1 (N) ) = 0 or the previous objects are defined for every i ∈ N, in which case we let k := +∞.
Indeed, assume by contradiction that there is a first index i < k such that for some j 0 < i, (dist(x i n , x j0 n )) n∈σi(N) is not divergent.In particular, there exists σ ⪯ σ i such that → ∞, for every j < i, j ̸ = j 0 by minimality of i and the triangle inequality dist( Notice by (3.5) that for every n ∈ σ(N), and passing to the weak limit, knowing that x j0 n − x i n → −x and dist(x j n , x i n ) → +∞ for j 0 < j < i, This contradicts the fact that (τ and proves (3.8).
-By Lemma 3.3, it suffices to prove that m((µ v n ) n∈σ(N) ) = 0. We claim that: Let us show (3.13).Let σ ⪯ σ and (x n ) n∈σ(N) be a sequence of points such that We need to prove that ∥µ∥ ⩽ m((µ i n ) n∈σi(N) ) for every i < k.Assume without loss of generality that ∥µ∥ > 0. Then for every i < k, Otherwise, up to subsequence, (dist(x n , x i n )) n would be bounded by some constant M , and for every r > 0, for n large enough by (C).Hence µ would be 0, a contradiction.Up to further extraction, one can assume that (τ −xn µ n ) n∈σ(N) converges weakly to a measure µ ∈ M (R N ).Since µ v n ⩽ µ n , we have µ ⩽ µ.Moreover by (3.5), for every i < k and n ∈ σ(N) large enough, and because of (3.14) the sum converges weakly to 0, so that τ −xn µ i n , which is what had to be proved.
Step 6: re-centering of the bubbles at points of supp µ n .-By (3.10), (τ −x i n µ n ) n∈σ(N) converges weakly to the non-trivial measure µ i for every i < k, thus Therefore, for every n large enough, there is a point x i n such that |x i n − x i n | < R i and x i n ∈ supp µ n .After a further extraction, one may assume that for every i, where diam B i n = 2r i n for every n, and After replacing the balls B i n by B i n , (B) and (C) are satisfied by definition.Notice that (τ − x i n µ n ) n∈σ(N) converges weakly to µ i := τ pi µ i with ∥ µ i ∥ = ∥µ i ∥, and lim sup and (E) holds as well.□ Remark 3.4.-If the sequence of families of balls (B i n ) 0⩽i<kn satisfies the conclusion of the theorem, i.e., (A)-(E), then it is also the case for any family of balls ( B i n ) 0⩽i<kn with the same centers as those of B i n and with smaller but still divergent radii (i.e., satisfying (C)).It can be easily seen following the arguments at Step 6 of the proof.

3.2.
Lower bound by concentration-compactness. -We will first establish a lower bound for the minimal energy along vanishing sequences defined on varying subsets of R N .We say that a sequence of Borel functions (u n ) n∈N , each defined on some open set |u| < +∞.
It will be convenient to first extend our Sobolev functions to a neighbourhood Ω δ of Ω where for every δ > 0 and every set X ⊆ R N , we have set We will need to consider sufficiently regular domains for which we have an extension operator W 1,p ∩ L 1 uloc (Ω) → W 1,p ∩ L 1 uloc (Ω δ ).We will only apply it to domains with smooth boundary, in which case we can use a reflection technique.Since we want quantitative estimates, we will use the notion of reach of a set X ⊆ R N (see [Fed59]).We say that X has positive reach if there exists δ > 0 such that every x ∈ X δ has a unique nearest point π(x) on X.The greatest δ for which this holds is denoted by reach(X) and the map x ∈ X reach(X) → π(x) ∈ X is called the nearest point retraction.
Moreover, by the change of variable formula and the chain rule, u satisfies the desired estimates since σ is bi-Lipschitz with its Lipschitz constants controlled in terms of δ and reach(∂Ω).□ We will need a localized version of the Gagliardo-Nirenberg-Sobolev inequality in a particular case: Lemma 3.7.-Let Ω ⊆ R N be an open set such that ∂Ω is C 1 with positive reach, let p ∈ [1, +∞), let r ⩾ p(1 + 1/N ), and assume that r ⩽ pN /(N − p) when p < N .Then for every u ∈ L 1 ∩ W 1,p (Ω), , where α ∈ (0, 1] is the unique parameter such that 1/r = α(1/p − 1/N ) + (1 − α), and the constant C < +∞ depends on N, r, p and reach(∂Ω).
Proof of Proposition 3.8.-Suppose for example that u n ⩾ 0 a.e. for every n.Without loss of generality, we may assume after extracting a subsequence that: (3.18) For all n ∈ N, we consider the measure ν n ∈ M + (R N × R × R N ) defined as the pushforward of the probability measure µ n = 1 θn u n L N Ω n by the map (Φ n , u n , ∇u n ), that is: We are going to show in several steps that ν n and deduce the result.It suffices to show that the three projections ν i n := (π i ) ♯ ν n , i ∈ {1, 2, 3} converge narrowly to δ x0 , δ 0 and δ 0 respectively.Indeed, this would imply that (ν n ) converges narrowly to a measure concentrated on (x 0 , 0, 0), hence to δ (x0,0,0) since the ν n are probability measures.First of all, since (ν n ) has bounded mass and (θ n ) is bounded, we may take a subsequence (not relabeled) such that ν n Step 1: Step 2: -By (3.18) and our assumption (H 5 ), there is a constant J.É.P. -M., 2024, tome 11 We deduce from Markov's inequality, and Lemma 3.7 applied with r = p(1 + 1/N ), corresponding to α = N/(N + 1), that , where in the last inequality, we have used the identity αr = p and (3.19), and C, C ′ depend only on N, r, p and inf n reach(∂Ω n ).
Since (u n ) n∈N is vanishing and (θ n ) n∈N is bounded, the last term in the previous inequality goes to zero as n → ∞ and it follows that Step 3: -Fix M > 0 and η > 0. One has by (3.19), By the previous step, we know that lim n→+∞ ν 2 n ([η, +∞)) = 0, hence taking the superior limit as n → +∞ then η → 0 we get lim n→+∞ ν 3 n ([M, +∞)) = 0. Since this is true for every M > 0 we obtain Step 4: conclusion.-By the previous steps, we deduce that ν n and by definition of f ′ − (see (2.10)), we have g(x, 0, 0) = f ′ − (x, 0 + , 0) for every x ∈ R N .Hence, by lower semicontinuity of g and weak convergence of (ν n ), we get which ends the proof of the lemma.□ As a corollary, we may now relate the slope at 0 of H f to that of f .Corollary 3.9.-Assume that f : , and the converse inequality ) is a sequence of functions of mass θ n = ´RN u n going to 0 and which is almost minimizing in the sense that We now establish our main energy lower bound along sequences with bounded mass (not necessarily vanishing): loc (B Rn , R ± ) with finite limit mass m := lim n→∞ ´Brn u n , and Then there exists a family (u i ) 0⩽i<k of functions in W 1,1 loc (R N , R ± ) with k ∈ N∪{+∞}, such that m i := ´RN u i ∈ R * ± for every i, and Proof.-Suppose for example that u n ⩾ 0 a.e. for every n.We first assume, up to subsequence, that the left hand side of (3.22) is a finite limit.We apply the profile decomposition Theorem 3.2 to the sequence of positive measures µ n = u n L N B rn where, we assume the extraction σ to be the identity for convenience, and we use the same notation as in Theorem 3.2.In particular, for each bubble By assumption, we have lim n→∞ (R n −r n ) = +∞; hence, up to reducing the radii of the balls B i n if necessary, in such a way that their radii still diverge (see Remark 3.4), we can assume that ) is assumed to be finite, we get that the sequence (u i n ) n is bounded in W 1,p loc (R N ) by (H 5 ).Hence, after a further extraction if needed, we get that (u i n ) n∈N ⇀ u i weakly in W 1,p loc (R N ) for some limit u i , for every 0 ⩽ i < k = lim k n .Setting m i = ´RN u i for every i, by (D) in Theorem 3.2, we have Fix ε > 0. We decompose the energy as Note that the domains Ω n := B Rn ∖ 0⩽i<k B i n satisfy inf n∈N reach(∂Ω n ) > 0 as noticed in Example 3.5, thanks to (3.23) and (B), (C) in Theorem 3.2.Hence, applying Proposition 3.8 to the Lagrangian f ε , we obtain Moreover, by the lower semicontinuity of integral functionals (see [But89, Th. 4.1.1]),in view of (3.20), we have for each i with 0 ⩽ i < k, Finally, by (3.24), (3.25), (3.26), (H 8 ) together with monotone convergence, we deduce that The similar statement for non-positive functions is obtained in the same way.□ 3.3.Existence of optimal profiles.-For the existence of an optimal profile in (1.2), we need a criterion that rules out splitting and vanishing of minimizing sequences: Lemma 3.11.-Let H : R + → R + be a concave function.Then H is subadditive, and if for some 0 < θ < m one has H(m) = H(m − θ) + H(θ), then H is linear on (0, m).
Proof.-By concavity, t → H(t)/t is non-increasing.Hence, But, by assumption, the last inequality is an equality which means that We can now state and prove our existence result: Proof.-We consider the case m ⩾ 0, the case m < 0 can then be deduced by considering f (u, ξ) = f (−u, −ξ).We assume without loss of generality that H f is finite on (0, +∞), otherwise by Theorem 1.1 there is nothing to prove.By Remark 2.1, the admissible class in (1.3) can be reduced to non-negative functions.In particular, if m = 0, then u = 0 is the only non-negative solution.If m > 0, we apply Proposition 3.10 in the following situation: is a minimizing sequence for the minimization problem in (1.3), and (r n ) n∈N is a sequence of positive radii going to +∞ such that lim n→∞ ´Brn u n = m.We obtain , where m i := ´RN u i .By Proposition 2.5 and Proposition 2.6, in view of our assumption (H 6 ), and since H f is assumed to be finite on (0, +∞) (for the case N = 1), we have Since the concave function H f is not linear on [0, m], by Lemma 3.11, we have either k = 1 and m v = 0, and we are done, or k = 0 and m = m v .But in the latter case, we would have . This contradicts our assumption.□ Remark 3.13.-Notice that the end of the proof actually shows, under the given assumptions, that the set of minimizers for a given mass m is compact in L 1 modulo translations.

Γ-convergence of the rescaled energies towards the H-mass
We establish lower and upper bounds for the Γ-lim inf and Γ-lim sup respectively, from which we deduce the proof of our main Γ-convergence result.The upper bound on the Γ-lim sup holds under more general assumptions and will be needed in Section 5.5.
), with the usual convention (±∞) × 0 = 0. Notice that it is concave on R + and R − by Theorem 1.1, and under (H 6 ) we have , (H 2 ), (H 3 ), (H 5 ) and (H 8 ) where f = lim ε→0 f ε .Let (ε n ) n∈N be a sequence of positive numbers going to zero, (u n ) n∈N be a sequence in W 1,1 loc (R N ), and let be the energy measure associated with In particular, Proof of Proposition 4.1.-Set H := H − f and recall that it is concave on R + and R − by Theorem 1.1.Let us assume first that u n ⩾ 0 a.e. for every n.To obtain (4.2), it is enough to prove that for every x 0 ∈ R N , (4.3) e({x 0 }) ⩾ H(x 0 , µ({x 0 })).
Step 4: proof of (4.2) for signed (u n ) n .-Notice that the preceding reasoning for nonnegative u n applies also to the case of non-positive u n .Let us handle the case where the (u n )'s may change sign.We simply apply the above cases to the positive and negative parts ((u n ) ± ) n which converge weakly as measures (up to subsequence) to some measures µ ± ∈ M + (R N ) which satisfy µ = µ + − µ − , so that e ⩾ H(±µ ± ).
Step 5: lower bound for the Γ-lim inf.-We justify that (4.2) implies the lower bound Γ(C ′ 0 )-lim inf ε→0 E ε ⩾ M H . Indeed, fix µ ∈ M (R N ) and consider a family (u ε ) ε>0 weakly converging to µ as ε → 0. We need to show that M H (µ) ⩽ lim inf ε→0 E ε (u ε ).Assume without loss of generality that the inferior limit is finite and take a sequence of positive numbers (ε n ) n∈N → 0 such that this inferior limit is equal to lim n→∞ E εn (u εn ).Now the energy measure e n associated with u n = u εn has bounded mass and up to extracting a subsequence one may assume that it converges weakly to some measure e ∈ M + (R N ).By the previous steps, e ⩾ H(µ), and by lower semicontinuity and monotonicity of the mass: where H f,lsc ⩽ H f stands for the lower semicontinuous envelope of H f , defined in (2.5).In other words, we have As an upper Γ-limit, F is sequentially lower semicontinuous in the narrow topology.Hence, by Proposition 2.4, it is enough to prove that and assume without loss of generality that x i ̸ = x j when i ̸ = j and M H f (µ) < +∞.Fix η > 0. For each i = 1, . . ., k, there exists which converge narrowly as measures to u as ε → 0. We have by change of variables: Using our assumption (U) and the dominated convergence theorem, one gets as ε → 0: Our assumption (H 6 ) is not very standard, but we need a condition of this type in order to get Γ-convergence of the rescaled energies E ε towards M H f , as shown by the following class of scale-invariant Lagrangians: (5.1) where p ∈ (1, N ), N ∈ N * and p ⋆ := pN /(N − p).By straightforward computations, E ε (u) = E f (u) := ´RN f (u, ∇u) for every ε > 0 and u ∈ W 1,p loc (R N ) in that case.Moreover, the associated minimal cost function H f is not trivial.Indeed, applying the Gagliardo-Nirenberg-Sobolev inequality, Hence, for every m > 0, we have H f (m) > 0, and even H f (m) < +∞ since any function u = v p ⋆ , with v ∈ W 1,p (R N , R + ), has finite energy.Replacing u by mu in the infimum defining H f in (1.2), we actually obtain In that case, it is clear that the Γ-limit of E ε ≡ E in the weak or narrow topology of M + (R N ), that is the lower semicontinuous relaxation of E f , does not coincide with M H f ; indeed, the first functional is finite on diffuse measures whose density has finite energy, while the second functional is always infinite for non-trivial diffuse measures since H ′ f (0 + ) = +∞.These scaling invariant Lagrangians are ruled out by our assumption (H 6 ).All the other assumptions are satisfied except (H 5 ).Note that the following perturbation of f , f (u, ξ) = 1 + u p(1/p ⋆ −1) |ξ| p satisfies all the assumptions except (H 6 ), and provides a counterexample to the Γ-convergence.Indeed, M H f ⩾ M H f is still infinite on diffuse measures, while (the relaxation of) E f is finite for any diffuse measure whose density has finite energy.
We stress that an assumption like (H 6 ) is actually needed, even for the lower semicontinuity of the function H f -recall that if M H f is a Γ-limit, then it must be lower (8) Actually, we apply it to vε = ϕε(u) where ϕε is a suitable approximation of (•) 1/p ⋆ and take ε → 0.
semicontinuous by [Bra02, Prop.1.28], which in turn implies that the function H f is lower semicontinuous by Proposition 2.4.Indeed, consider the Lagrangians as can be easily seen via the change of function ε N u(ε •), with ε > 0 small.5.2.General concave costs in dimension one.-It has been proved in [Wir19] that for any continuous concave function H : R + → R + with H(0) = 0, there exists a function c : R + → R + such that c(0) = 0, u → c(u)/u is lower semicontinuous and non-increasing on (0, +∞), and for every m ⩾ 0, The Lagrangians of the form f ε (x, u, ξ) = |ξ| 2 + c(u), in dimension N = 1, satisfy all our assumptions (H 1 )-(H 8 ), hence our Γ-convergence result stated in Theorem 1.2 yields the Γ-convergence of the functionals towards M H for both the weak and narrow convergence of measures.Therefore, we may find an elliptic approximation of any concave H-mass.Let us stress that c is determined in [Wir19] from H through several operations including a deconvolution problem, but no closed form solution is given in general; nonetheless, an explicit solution is provided if c is affine by parts.In higher dimension N ⩾ 2, Proposition 2.8 tells us that the class of functions H = H f with f satisfying (H 1 )-(H 8 ) is smaller, namely, H must satisfy: H is strictly concave (m * , +∞).
We have no positive or negative answer to the inverse problem, consisting in finding f satisfying our assumptions such that H f = H, for a given continuous concave function H : R + → R + satisfying (5.3).

Homogeneous costs in any dimension.
-In this section, we provide Lagrangians f to obtain the α-mass M α := M t →t α in any dimension N for a wide range of exponents, including exponents α ∈ 1 − 1 N , 1 .We consider for every p ∈ [1, +∞), s ∈ (−∞, 1] and N ∈ N * , the energy defined for every u Notice that for p > 1, f satisfies all our hypotheses (H gives a uniform integrable decay in |x| −N/s at infinity ; when s ⩽ 0, we use the elementary inequality t + t s ⩾ 1 for all t > 0, to obtain (5.9) so that the size of the support of u n (a ball of radius R n > 0) is uniformly bounded with respect to n.We deduce by lower semicontinuity of E f , that, extracting a subsequence if necessary, (u n ) n converges in L 1 to a minimizer u of E f with mass m, i.e., E f (u) = H f (m) and ´RN u = m.In particular, H f (1) > 0. In other words, u solves in the weak sense the Euler-Lagrange equation where λ ∈ R is the Lagrange multiplier associated to the mass constraint.One can see that λ = H ′ f (m) = αm α−1 H f (1).Indeed, let u 1 be a minimizer of E f of mass 1 ; then a minimizer of E f with mass m is given by u m (•) := mλ N m u 1 (λ m •), with λ m = m s−p/(N p+p−N s) (see (5.5)) ; hence, In particular, λ is unique.
From the Euler-Lagrange equation, we also get that u is smooth in B(0, R) by a bootstrap argument ; hence, u solves the Euler-Lagrange equation in the classical sense.In terms of the profile v, the Euler-Lagrange equation rewrites (5.12) For every r ∈ (0, R), integrating (5.12) on (0, r) yields The LHS in (5.13) is non-positive as v ′ ⩽ 0 in (0, R).If s > 0 and R = +∞, the RHS is +∞ since the integrand goes to +∞ as ρ → R, which is a contradiction.When s ⩽ 0, we already saw in (5.9) that the support of v is bounded.In any case, we have thus proved that R < +∞.

Branched transport approximation: H-masses of normal 1-currents
Branched transport is a variant of classical optimal transport (see [San15] and Section 4.4.2therein for a brief presentation of branched transport, and [BCM09] for a vast exposition) where the transport energy concentrates on a network, i.e., a 1-dimensional subset of R d , which has a graph structure when optimized with prescribed source and target measures.It can be formulated as a minimal flow problem, min M H 1 (w) : div(w) = µ − − µ + , where µ ± are probability measures on R d , H : R d × R + → R + is mass-concave, and the H-mass M H 1 is this time defined for finite vector measures w ∈ M (R d , R d ) whose distributional divergence is also a finite measure; in the language of currents, it is called a 1-dimensional normal current.Any such measure may be decomposed into a 1-rectifiable part θξ • H 1 Σ where θ(x) ⩾ 0 and ξ(x) is a unit tangent vector to Σ for H 1 -a.e.x ∈ Σ, and a 1-diffuse part w ⊥ satisfying |w ⊥ |(A) = 0 for every 1-rectifiable set A: w = θξ • H 1 M + w ⊥ .
In the case H(x, m) = m α with 0 < α < 1, a family of approximations of these functional has been introduced in [OS11]: (5.15) The functionals E ε Γ-converge as ε → 0 + , in the narrow topology, to cM α for some non-trivial c, as shown in Section 5.3, and one may recover every α-mass in this way for α ∈ ((2d − 4)/(2d + 1), 1], and in particular every so-called super-critical exponents for branched transport in dimension d, that is α ∈ (1 − 1/d, 1]. The same slicing method would allow to extend our Γ-convergence result stated in Theorem 1.2 to functionals defined on vector measure (5.16) for Lagrangians f ε → f fitting the framework of Theorem 1.2.The expected Γ-limit, for the weak topology of measures and their divergence measure, would be the functional M H f 1 , with H f defined in (1.2).Note that this approach would provide approximations of H-masses for more general continuous and concave cost functions H : R + → R + satisfying H(0) = 0.By [Wir19], we would obtain all such H-masses when N = 1 (corresponding to d = 2).

NotationB
r (x) open ball of radius r centered at x; B r open ball B r (0); M (R N ) set of finite signed Borel measures on R N; M + (R N ) set of finite positive Borel measures on R N ; Φ ♯ µ pushforward of a measure µ ∈ M (R N ) by a map Φ : R N → R k , defined as A → µ(Φ −1 (A)); τ x µ Borel measure A → µ(A − x) if µ ∈ M (R N ) and x ∈ R N ; c B µ Borel measure τ −x (µ B) if B is the ball B r (x);µ ℓ C ′ 0 − − ⇀ µ weak convergence of measures, i.e., weak-⋆ convergence in duality with the space C 0 (R N ) of continuous functions vanishing at infinity; µ ℓ C ′ b − − ⇀ µ narrow convergence of measures, i.e., weak-⋆ convergence in duality with the space C b (R N ) of continuous and bounded function; Σ set of increasing maps σ : N→ N; σ 1 ⪯ σ 2 σ 1 , σ 2 ∈ Σ are such that σ 1 ( n, +∞ ) ⊆ σ 2 (N)for some n ∈ N ; ± fixed to either + or − in the whole statement or proof, and ∓ = −(±).
β = (2 − 2d + 2αd)/(3 − d + α(d − 1)), γ 1 = (d − 1)(1 − α) and γ 2 = 3 − d + α(d − 1).It has been shown in[OS11,Mon17] that the functionals E ε Γ-converge as ε → 0 + , in the topology of weak convergence of u and its divergence, to a non-trivial multiple of the α-massM α 1 := M H 1 with H(x, m) = m α in dimension d = 2.The result extends to any dimension d, by[Mon15], thanks to a slicing method that relates the energy E ε with the energy of the sliced measures u = (w • ν) + supported on the slices V a = {x ∈ R d : x • ν = a} ≃ R N , for any given unit vector ν ∈ R d , defined by If t * < +∞, we have u ∧ u t * = 0 a.e. and u t → u t * locally in measure as t → t * by continuity of translation in L 1 .Thus using dominated convergence again, Proof of Proposition 2.4.-Since H lsc is lower semicontinuous and mass-concave, we know from Proposition 2.3 that M H lsc is sequentially lower semicontinuous in the weak topology hence also in the narrow topology of M (R N ).Since M H lsc ⩽ M H atom , we deduce that M H lsc is lower or equal than the sequentially lower semicontinuous envelope of M H atom in the narrow topology, i.e., the right hand side in (2.6), which we denote by J.É.P. -M., 2024, tome 11 function.Nevertheless the subadditivity in the mass m and the lower semicontinuity would be necessary if H did not depend on x.
and we set t n+1 = t n +δ 2 n .By Lemma 2.9, v n ∈ M f In particular, the monotone function t → H(t)/t must be constant on [θ, m], i.e., H must be linear on [θ, m].By concavity this is only possible if H is linear on [0, m].□ ′ b )-lim sup ε→0 E ε ⩽ M H f,lsc from Proposition 4.2, where the assumption (U) is a consequence of (H 4 ), (H 7 ) and (H 8 ).By (H 6 ) and Theorem 1.1, H − f = H f , and since H f ⩾ H f,lsc by definition, both Γ-lim inf and Γ-lim sup (for weak and narrow topologies) coincide with M H f .□