Regularity of minimizers for a model of charged droplets

TL;DR

This paper studies the regularity of minimizers in a variational model for charged liquid droplets, incorporating screening effects, and establishes partial regularity results to advance understanding of droplet shapes.

Contribution

It provides the first partial regularity results for minimizers in a charged droplet model considering screening effects, enhancing theoretical understanding.

Findings

Proved partial regularity of minimizers.

Incorporated screening effects into the regularity analysis.

Laid groundwork for further analysis of droplet shapes.

Abstract

We investigate properties of minimizers of a variational model describing the shape of charged liquid droplets. The model, proposed by Muratov and Novaga, takes into account the regularizing effect due to the screening of free counterionions in the droplet. In particular we prove partial regularity of minimizers, a first step toward the understanding of further properties of minimizers.

Equations

$$P(E)+\frac{Q^2}{\mathcal{C}(E)}.$$

$$\frac{1}{\mathcal{C}(E)}:=\inf\Bigg{\{}\frac{1}{4\pi}\iint\frac{d\mu(x)d\mu(y)}{|x-y|}:\operatorname{spt}\mu\subset E,\mu(E)=1\Bigg{\}},$$

$$\underset{|E|=V}{\min }P(E)+\frac{Q^2}{\mathcal{C}(E)}.$$

$$\underset{|E|=V}{\inf }P(E)+\frac{Q^2}{\mathcal{C}(E)}=P(B^V),$$

$$\mathcal{F}(E,u,\rho):=P(E)+Q^{2}\Bigg{\{}\int_{\mathbb{R}^{n}}a_{E}|\nabla{u}|^{2}\,dx+K\int_{E}\rho^{2}\,dx\Bigg{\}}.$$

$$a_E(x):=1_{E^c}+\beta 1_E,$$

$$\rho 1_{E^c}=0\text{and}\int \rho =1,$$

$$-\operatorname{div}\big{(}a_{E}\,\nabla{u}\big{)}=\rho\qquad\text{in}\;\mathcal{D}^{\prime}(\mathbb{R}^{n}).$$

$$\sigma P(E)+Q^{2}\Bigg{\{}\frac{\beta_{0}}{2}\int_{\mathbb{R}^{n}}a_{E}|\nabla{u}|^{2}\,dx+K\int_{E}\rho^{2}\,dx\Bigg{\}}$$

$$\min\big{\{}\mathcal{F}(E,u,\rho):|E|=V,E\subset B_{R},(u,\rho)\in\mathcal{A}(E)\big{\}},$$

$$\mathcal{A}(E):=\big{\{}\text{$(u,\rho)\in D^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})$: $u$ and $\rho$ satisfy\leavevmode\nobreak\ \eqref{vincolo1} and\leavevmode\nobreak\ \eqref{e:rho}}\big{\}},$$

$$D^1({\mathbb{R}}^n)={\overline{C_c^{\infty }({\mathbb{R}}^n)}}^{{\mathring{W}}^{1,2}({\mathbb{R}}^n)}\Vert \phi {\Vert }_{{\mathring{W}}^{1,2}({\mathbb{R}}^n)}=\Vert \nabla \phi {\Vert }_{L^2({\mathbb{R}}^n)}.$$

$$-\Delta u=\rho ,$$

$$\min_{|E|=V,E\subset B_{R}}P(E)+Q^{2}\min\Biggl{\{}\frac{1}{4\pi}\iint\frac{\rho(x)\rho(y)dxdy}{|x-y|}+K\int\rho^{2}\text{ s.t. $\rho\mathbf{1}_{E^{c}}=0,\int\rho=1$}\Biggr{\}},$$

$${\mathcal{G}}_{\beta ,K}(E):=\underset{(u,\rho )\in \mathcal{A}(E)}{\inf }\left\{{\int }_{{\mathbb{R}}^n}{a}_E|\nabla u{|}^2+K{\int }_E{\rho }^2\right\},$$

$$(u,\rho)\in\mathcal{A}(E)\quad\Longrightarrow\quad\Bigl{(}\lambda^{2-n}u\Bigl{(}\frac{\cdot}{\lambda}\Bigr{)},\lambda^{-n}\rho\Bigl{(}\frac{\cdot}{\lambda}\Bigr{)}\Bigr{)}\in\mathcal{A}(\lambda E),$$

$${\mathcal{G}}_{\beta ,{\lambda }^{2}K}(\lambda E)={\lambda }^{2-n}{\mathcal{G}}_{\beta ,K}(E).$$

$${\mathcal{F}}_{\beta ,K,Q}(E):=P(E)+Q^2{\mathcal{G}}_{\beta ,K}(E),$$

$${\mathcal{F}}_{\beta ,K,Q}(E)={\lambda }^{1-n}{\mathcal{F}}_{\beta ,K{\lambda }^2,Q{\lambda }^{\frac{2n-3}{2}}}(\lambda E).$$

$${\mathcal{P}}_{\beta ,K,Q,R}$$

$${\mathbf{e}}_E(x,r):=\underset{\nu \in {\mathbb{S}}^{n-1}}{\inf }\frac{1}{r^{n-1}}{\int }_{{\partial }^{\ast }E\cap B_r(x)}\frac{|{\nu }_E(y)-\nu {|}^2}{2}d{\mathcal{H}}^{n-1}(y),$$

$$D_E(x,r):=\frac{1}{r^{n-1}}{\int }_{B_r(x)}|\nabla u_E{|}^2,$$

$$r+{\mathbf{e}}_E(x,r)+D_E(x,r)\le {\epsilon }_{\text{reg}},$$

$$\mathrm{Id}\le A(x)\le \beta \mathrm{Id}\text{for all x∈Rn.}$$

$$-\mathrm{div}(A\nabla u)=\rho .$$

$$\Vert u{\Vert }_{2^{\ast }}\le S(n)\Vert \nabla u{\Vert }_{L^2}.$$

$$\mathcal{E}(v)=\frac{1}{2}\int A\nabla v\cdot \nabla v-\int \rho v,$$

$$\underset{v\in D^1({\mathbb{R}}^n)}{\min }\mathcal{E}(v)=\mathcal{E}(u)=-\frac{1}{2}\int A\nabla u\nabla u=-\frac{1}{2}\int \rho u.$$

$$-\Delta u=\rho ,$$

$$\rho =\frac{1_E}{|E|},-\mathrm{div}(a_E\nabla u)=\rho ,$$

Full text

1. Introduction

1.1. Background and description of the model

In this paper we investigate the regularity of minimizers for a variational model describing the shape of charged liquid droplets. Roughly speaking, the shape of a charged liquid droplet is determined by the competition between an “aggregating” term, due to surface tension forces, and to a “disaggregating” term due to the repulsion effect between charged particles.

Several models proposed in literature are based on this principle. Among them, one of the simplest and most used assumes that charged droplets are stationary points for the following free energy:

[TABLE]

Here, $E\subset\mathbb{R}^{3}$ corresponds to the volume occupied by the droplet, $P(E)$ is its perimeter, $Q$ is the total charge and

[TABLE]

takes into account the repulsive forces between charged particles. Note that $\mu$ can be though as a (normalized) density of charges and that $\mathcal{C}(E)$ is the classical Newtonian capacity of the set $E$. In particular one assumes that the optimal shapes are given by the following variational problem:

[TABLE]

Heuristically, one expects the perimeter term to dominate for small value of the charge $Q$ thus forcing the droplet to have a spherical or almost spherical shape, while the repulsion term should become dominant for large values of $Q$, thus leading to the formation of singularities and/or to the ill-posedness of (1.3). This heuristics is confirmed by the perturbative analysis of (1.1) around a spherical shape. This computation, performed for the first time by Lord Rayleigh in 1882,  [22], shows that the spherical droplet is linearly stable only for $Q$ smaller than a critical threshold. This is known as the Rayleigh criterion.

The transition from a stable to an unstable behavior of spherical droplets has also been verified experimentally, starting from the work of Zeleny at the beginning of 1900 [29] (in a slightly different context). More precisely, it has been observed that a spherical droplet exposed to an electric field, remains stable until the total charge is below a critical value $Q_{c}>0$, while, as soon as $Q$ exceeds $Q_{c}$ the droplet changes its appearance and the surface start to develop singularities, the so called Taylor’s cones, [26]. Whenever $Q\geq Q_{c}$ a very thin steady jet composed by small but highly charged little balls is formed, [28, 9, 23, 10].

In spite of the interest of (1.3) in applications, a rigorous mathematical study of this model has been only performed in the last years, mostly thanks to the work of Goldman, Muratov, Novaga and Ruffini, see [15, 20, 17, 21, 16] and references therein.

The starting point of their analysis is the following remarkable and somehow disappointing observation: Problem (1.3) is always ill-posed. More precisely, in [15], it is shown that

[TABLE]

where $B^{V}$ is the ball of volume $V$. Since $B^{V}$ is a competitor for the variational problem , this clearly implies that there are no minimizers of (1.3).

The above equality is obtained by constructing a minimizing sequence consisting of a ball of roughly volume $V$ together with several balls with vanishing perimeter and volumes and very high charge escaping at infinity. Hence, on the mathematical side, the phenomena observed by Zeleny appears for every value of the charge. Let us also remark that ill-posedness of (1.3) is shown also if one assumes that all the set involved in the minimization problem are a-priori bounded, [15, Theorem 1.3].

It then becomes natural to investigate the local minimality of the ball, at least for “small” perturbations and small values of $Q$. In [15, Theorems 1.4 & 1.7] the linear stability of the ball in the small charge regime, is upgraded to local minimality in a sufficiently strong topology. On the other hand Muratov and Novaga showed that the ball is never a local minimizer of (1.3) under (smooth) perturbation which are small in $L^{\infty}$, [20, Theorem 2]. We also refer the reader to [17] where well-posedness is recovered under suitable geometric restrictions and to [21] for the case of “flat” droplets.

The main phenomena driving to the ill-posedness of (1.3) is the possibility of concentrating a high charge on small volumes. In order to avoid this situation, in [20], Muratov and Novaga proposed as a possible regularization mechanism the finite screening length in the conducting liquid , by introducing the entropic effects associated with the presence of free ions in the liquid, see also [8, 27] for a related model. They suggested to consider the following Debye-Hückel-type free energy (in every dimension)

[TABLE]

Here

[TABLE]

where $\mathbf{1}_{F}$ is the characteristic function of a set $F$ and $\beta>1$ is the permittivity of the droplet. The (normalized) density of charge $\rho\in L^{2}(\mathbb{R}^{n})$ satisfies

[TABLE]

and the electrostatic potential $u$ is such that $\nabla u\in L^{2}(\mathbb{R}^{n})$ and

[TABLE]

$K>0$ is a physical constant related to the model.111Actually in [20], the energy (1.4) is written as

\sigma P(E)+Q^{2}\Bigg{\{}\frac{\beta_{0}}{2}\int_{\mathbb{R}^{n}}a_{E}|\nabla{u}|^{2}\,dx+K\int_{E}\rho^{2}\,dx\Bigg{\}}

for suitable parameters $\sigma$ and $\beta_{0}$ and the relation (1.6) is replaced by -\beta_{0}\operatorname{div}\big{(}a_{E}\,\nabla{u}\big{)}=\rho. However it is easy to see that the parameters $\sigma$ and $\beta_{0}$ can be absorbed in $Q$ and $K$, see also the discussion below.

The variational model proposed in [20], where one assumes a-priori that all the sets are contained in a fixed (large) ball $B_{R}$, is the following

[TABLE]

where we have set

[TABLE]

and

[TABLE]

Note that the class of admissible couples $\mathcal{A}(E)$ is non-empty only if $n\geq 3$, for this reason this assumption will be in force throughout the whole paper, see also Remark 2.2.

Thanks to the a-priori boundedness assumption $E\subset B_{R}$, existence of a minimizer in the class of sets of finite perimeter can be easily shown, see [20, Theorem 3].

Note that the presence of the $L^{2}$ norm of $\rho$ in the energy is exactly what prevents the concentration of charges. Indeed, if one assumes that $\beta=1$ so that (1.6) reduces to

[TABLE]

then the minimization problem (1.7) can be written, in dimension $n=3$ as

[TABLE]

which should be compared with (1.2) and (1.3). In view of this we also note that, on the mathematical ground, the variational problem (1.7) can also be considered as an “interpolation” between the classical Otha-Kawasaki problem, and the free-interfaces problems arising in optimal design studied for instance in [3, 18, 6, 12].

1.2. Main results

Once existence of a minimizers of (1.7) is obtained it is natural to investigate their qualitative and quantitative properties, also to understand to which extent the predictions of model agree with the observed phenomenology. In particular the following questions arise, compare with [20]:

Is every minimizers smooth, at least outside a small singular set? 2. -

Which is the structure of (possible) singularities of minimizers? Do they agree with Taylor’s cones222Note that this is possible only if $\beta$ is large compared to $1$, see the discussion at the end of this introduction and Remark 4.6? 3. -

Is it possible to show existence/non-existence of minimizers removing the a-priori confinement assumption? 4. -

Can one show that for small value of the charges minimizers of (1.7) are balls in agreement with experimental observations?

In this paper we address the question of regularity of minimizers. Our main result is the following partial regularity theorem:

Theorem 1.1**.**

Let $n\geq 3$ and $B>0$. Then there exists $\eta=\eta(n,B)>0$ with the following property: if $E$ is a minimizer of (1.7) with $\beta\leq B$ then there exists a closed set $\Sigma_{E}\subset\partial E$ such that $\mathcal{H}^{n-1-\eta}(\Sigma_{E})=0$ and $\partial E\setminus\Sigma_{E}$ is a $C^{1,\vartheta}$ manifold for all $\vartheta\in(0,1/2)$.

As it is customary in Geometric Measure Theory, the proof Theorem 1.1 is based on an $\varepsilon$-regularity result which is interesting on its own. In order to keep track of the various dependence on the parameters let us first fix some notations, which will be useful also in the sequel. For $E\subset\mathbb{R}^{n}$ we define

[TABLE]

where the set of admissible pairs $\mathcal{A}(E)$ is defined in (1.8) (if the dependence on the parameter is not relevant we will simply write $\mathcal{G}$). Since

[TABLE]

one has

[TABLE]

Setting

[TABLE]

one gets

[TABLE]

In particular, by replacing $K$ and $Q$ with $K(\omega_{n}/V)^{\frac{2}{n}}$ and $Q(\omega_{n}/V)^{1-\frac{3}{2n}}$ we can assume that $V=|B_{1}|=:\omega_{n}$. Namely, for $R\geq 1$ we will consider the following problem

[TABLE]

Furthermore, given a set of finite perimeter $E$ we define the spherical excess at a point $x$ and at scale $r>0$ as

[TABLE]

where, $\partial^{*}E$ is the reduced boundary of $E$, $\nu_{E}$ is the measure-theoretic unit normal to $\partial E$, see [19], and $B_{r}(x)$ is the ball of center $x$ and radius $r$. We also define the normalized Dirichlet energy as

[TABLE]

where $u_{E}$ is the minimizer in (1.9), whose existence and uniqueness can be easily proved, see Proposition 2.3 below. With these conventions, the $\varepsilon$-regularity results can be stated as follows, see also Theorem 8.1 below for a slightly more precise statement,

Theorem 1.2**.**

Given $n\geq 3$, $A>0$ and $\vartheta\in(0,1/2)$, there exits $\varepsilon_{\textnormal{reg}}=\varepsilon_{\textnormal{reg}}(n,A,\vartheta)>0$ such that if $E$ is minimizer of$\leavevmode\nobreak\ \eqref{e:problem}$ with $Q+\beta+K+\frac{1}{K}\leq A$, $x\in\partial E$ and

[TABLE]

then $E\cap B(x,r/2)$ coincides with the epi-graph of a $C^{1,\vartheta}$ function. In particular $\partial E\cap B(x,r/2)$ is a $C^{1,\vartheta}$ $(n-1)$-dimensional manifold.

Let us conclude this section with some remarks:

First beside its intrinsic interest, combining Theorem 1.2 with the analysis of the linearized energy around a ball one can show show that the balls uniquely minimize ($\mathcal{P}_{\beta,K,Q,R}$) for small value of $Q$. This will be addressed in a forthcoming paper.

Second we note that the dimension of the singular set in Theorem 1.1 depends only on the gap between the two permittivity constants and not on the other parameters appearing in the model. On the other hand the “regularity scale” in Theorem 1.2 depends on all the parameters involved. A similar fact has been observed in the context of free interfaces models in [6, 12].

Finally, it seems reasonable to expect that $C^{1,\vartheta}$ regularity of $\partial E$ can be upgraded to $C^{\infty}$ smoothness by some bootstrap argument. We leave this interesting question open.

1.3. Strategy of the proof and structure of the paper

Though the energy we are considering has a certain similarity with those studied in optimal design problems, the fact that the minimization problem in definition (1.9) is performed only among admissible pairs $(u,\rho)\in\mathcal{A}(E)$ makes very difficult to make local perturbations. In particular, problem ($\mathcal{P}_{\beta,K,Q,R}$) has (a priori) no local scaling invariance. For this reason in Section 2 we study carefully the energy $\mathcal{G}(E)$ and its minimizers $(u_{E},\rho_{E})$. Moreover we establish boundedness of $u_{E}$ and $\rho_{E}$ .

In order to study the regularity of minimizers one needs to perform local variations and hope that these gives localized (or almost localized) changes of the energy. This is not completely evident due to the presence of a volume constraint and to the non-local character of $\mathcal{G}$. As, it is well known, the volume constraint can be relaxed into a “perturbed” minimality property of minimizers. In order to have estimates uniform in the structural parameters it will be important to have this “perturbed” minimality property uniform in the class of minimizers. In Section 3 we start studying how the energy varies according to a flow of diffeomorphism, which will be important in performing small volume adjustments and we establish the Euler Lagrange equations for minimizers. In Section 4 we prove the perturbed minimality property and we study the behavior of the energy under local perturbations. In Section 5 we prove the compactness of the class of minimizers in the $L^{1}$ topology, which though not used in the proof of our main results is interesting by its own.

The next step consists in establishing local perimeter and volume estimates for the minimizers of ($\mathcal{P}_{\beta,K,Q,R}$). Usually these estimates are easily obtained by combining minimality with local isoperimetric inequalities. Here, due to the non-local character of the energy term $\mathcal{G}(E)$ and the absence of a natural scaling invariance of the problem, more refined arguments are required. In particular we will first show that the energy $\mathcal{G}$ is monotone by set inclusion. This implies that $E$ is an outer minimizer for the perimeter and leads to upper perimeter bounds and lower density estimates for $E^{c}$. Estimating the density of $E$ is instead more complicated and requires to perform an inductive argument showing that if $E$ has small relative measure in a ball $B_{r}(x)$, then the Dirichlet energy of $u_{E}$ decays enough to preserve this information at smaller scales, leading to a contradiction. In doing this, higher integrability of the gradient of minimizers of $\mathcal{G}$ plays a key role. Local density estimates are obtained in Section 6 together with the boundedness of $D_{E}(x,r)$. This fact combined with the local density and perimeter estimates allow somehow to recover the scaling invariance of the problem.

The main step of the proof of Theorems 1.1 and  1.2 is the decay of the excess established in Section 7. Once the local scaling invariance of the problem is recovered, the proof Theorem 1.2 follows the classical De Giorgi’s idea of harmonic approximation. Namely we will show that in the regime of small excess and small normalized Dirichlet energy, $\partial E$ can be well approximated by the graph of a function with “small” laplacian. This leads to the decay of the excess which, thanks to the higher integrability of $\nabla u_{E}$, in turn also implies the decay of the normalized Dirichlet energy and eventually allows to conclude the proof.

In Section 8 we prove Theorems 1.1 and 1.2. Theorem 1.2 will be an immediate consequence of Theorem 7.1 (see also Theorem 8.1 for a more quantitative version). Theorem 1.1 is proved by following the strategy of [12] where one combines the the $\varepsilon$-regularity result with the higher integrability of the $\nabla u_{E}$ and the classical regularity theory for minimal surfaces.

Let us remark that most of the above described difficulties arises only in the case when $\beta$ is relatively large compared to $1$. Indeed in the regime $\beta-1\ll 1$, Cordes estimates, see [4], imply that $\nabla u_{E}$ belongs to $L^{p}$ with $p$ large. In this case Hölder inequality immediately gives that the energy term $\mathcal{G}$ is lower order with respect to the perimeter at small scales. $E$ will then be an $\omega$-minimizer of the perimeter and the regularity theory follows for instance from [25], see Remark 4.6. In particular in this case one obtains full regularity in $n=3$, thus excluding the formation of Taylor’s cone singularities. This phenomena was already observed in [24] for a different model of charged droplets.

Acknowledgements

The work of G. D. P. and of G. V. is supported by the INDAM-grant “Geometric Variational Problems”.

2. Properties of minimizers of $\mathcal{G}$

In this section we start establishing some basic properties of minimizers of $\mathcal{G}$. We start with the following easy lemma. Here and in the following let $2^{*}:=2n/(n-2)$ (recall that we are always working with $n\geq 3$).

Lemma 2.1**.**

Let $n\geq 3$, $\beta>1$ and let $A:\mathbb{R}^{n}\to\operatorname{Sym}_{n}(\mathbb{R}^{n})$ be a symmetric matrix valued function such that

[TABLE]

Then for every $\rho\in L^{(2^{*})^{\prime}}$ (i.e. the dual of $L^{2^{*}}$) there exists an unique $u\in D^{1}(\mathbb{R}^{n})$ such that

[TABLE]

Proof.

Recall that for $u\in D^{1}(\mathbb{R}^{n})$ one has the following Sobolev inequality

[TABLE]

In particular by the assumptions on $\rho$ and $A$ the energy

[TABLE]

is finite. By Young’s inequality $\mathcal{E}(v)$ is bounded from below by $\|\nabla v\|_{L^{2}}^{2}-C(n)\|\rho\|_{(2^{*})^{\prime}}^{2}$. Direct methods of the calculus of variations imply the existence of a unique minimizers which is the desired solution. Furthermore for the solution we have

[TABLE]

∎

Remark 2.2**.**

In dimension $n=2$ the above lemma is easily seen to be false, indeed even for a smooth and compactly supported $\rho$, the solution of

[TABLE]

does not in general satisfy $\nabla u\in L^{2}$.

By the above lemma , if $|E|<\infty$ the couple $(u,\rho)$ defined by

[TABLE]

is admissible, $(u,\rho)\in\mathcal{A}(E)$. By testing the equation by $u$ and using the Sobolev embedding, we then get

[TABLE]

In particular (recall $\beta>1$)

[TABLE]

Proposition 2.3**.**

Let $E\subset\mathbb{R}^{n}$ be a set of finite measure. Then there exists an unique pair $(u_{E},\rho_{E})\in\mathcal{A}(E)$ minimizing $\mathcal{G}_{\beta,K}(E)$. Moreover

[TABLE]

and

[TABLE]

In particular $\rho_{E}\in L^{p}$ for all $p\in[1,\infty]$ and

[TABLE]

Proof.

Existence of a minimizer is an immediate application of the Direct methods in the Calculus of Variations. Uniqueness follows from the convexity of the admissible set $\mathcal{A}(E)$ and of the strict convexity of the energy

[TABLE]

Let now $\psi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that

[TABLE]

Let $v\in D^{1}(\mathbb{R}^{n})$ be the solution of

[TABLE]

If $(u_{E},\rho_{E})$ is the minimizing pair then $(v_{\varepsilon},\rho_{\varepsilon})=(u_{E}+\varepsilon v,\rho_{E}+\varepsilon\psi)\in\mathcal{A}(E)$ is admissible. Hence, by taking the derivative with respect to $\varepsilon$ of its energy we get

[TABLE]

Since this holds for all $\psi$ satisfying (2.5) we get that $u_{E}+K\rho_{E}=\textrm{const}$ in $E$. By multiplying this equation by $\rho_{E}$ and integrating over $E$ we infer that the constant shall be equal to $\mathcal{G}(E)$, and this proves (2.2). In particular $u_{E}$ solves

[TABLE]

By testing the above with $(\mathcal{G}(E)-u_{E})_{-}=-\min\{0,\mathcal{G}(E)-u_{E}\}$ we obtain

[TABLE]

which implies the second half of the first inequality in (2.3). Testing (2.7) with $u_{-}=-\min\{0,u\}$ we obtain the first half. The second inequality in (2.3) follows now from the first and (2.2). Inequality (2.4) follows from (2.1). ∎

We establish now the monotonicity of $\mathcal{G}$ with respect to set inclusion. We start from the following lemma.

Lemma 2.4**.**

Let $A$, $B:\mathbb{R}^{n}\to\operatorname{Sym}_{n}(\mathbb{R}^{n})$ two symmetric matrix valued functions such that $\operatorname{Id}\leq A(x)\leq B(x)$ for all $x\in\mathbb{R}^{n}$. If $\rho\in L^{(2^{*})^{\prime}}(\mathbb{R}^{n})$ and $u,v\in D^{1}(\mathbb{R}^{n})$ are the unique solutions of

[TABLE]

then

[TABLE]

In particular

[TABLE]

Proof.

Let $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$ be the following functionals defined on $D^{1}(\mathbb{R}^{n})$:

[TABLE]

Hence $\mathcal{E}_{A}(w)\leq{}\mathcal{E}_{B}(w)$ for every $w\in D^{1}(\mathbb{R}^{n})$. Since the solutions of (2.8) are minimizers of these energies, compare with Lemma 2.1, we have

[TABLE]

Then

[TABLE]

and

[TABLE]

concluding the proof. ∎

The following corollary is an immediate consequence of the above lemma.

Corollary 2.5**.**

Let $E\subset F\subset\mathbb{R}^{n}$ be two sets of finite measure. Then

[TABLE]

Proof.

Let $(u_{E},\rho_{E})$ be the optimal pair for $E$ and let $v$ be a solution of

[TABLE]

Then $(v,\rho_{E})$ is admissible in the minimization problem defining $\mathcal{G}_{\beta,K}(F)$, hence

[TABLE]

where the last inequality follow from Lemma 2.4. ∎

We conclude this section by proving the continuity of $\mathcal{G}$ under $L^{1}$ convergence. Recall that given two sets $E$ and $F$, $E\Delta F:=(E\cup F)\setminus(E\cap F)$ is their symmetric difference.

Proposition 2.6**.**

Let $\{E_{h}\}$ be a sequence of sets with $|E_{h}|=:V_{h}\to V>0$ and let $E$ be such that $|E_{h}\Delta E|\to 0$, so that in particular $|E|=V$. Assume that $\beta_{h}\to\beta$ and that $K_{h}\to K$. Then

[TABLE]

Moreover, $\nabla u_{E_{h}}$ and $\rho_{E_{h}}$ converge in $L^{2}$ to $\nabla u_{E}$ and $\rho_{E}$ respectively.

Proof.

Note that by (2.1)

[TABLE]

Thus

[TABLE]

Moreover

[TABLE]

In particular, if $(u_{h},\rho_{h})=(u_{E_{h}},\rho_{E_{h}})$ is the minimizing pair for $\mathcal{G}_{\beta_{h},K_{h}}(E_{h})$, then up to subsequence there exists $(u,\rho)$ such that

[TABLE]

Since $(u_{h},\rho_{h})$ are in $\mathcal{A}(E_{h})$, one immediately deduces that $(u,\rho)\in\mathcal{A}(E)$ and thus, by lower semicontinuity,

[TABLE]

To prove the opposite inequality we take $(u_{E},\rho_{E})$ to be the minimizing pair for $\mathcal{G}_{\beta,K}(E)$ and we define $(w_{h},\widetilde{\rho}_{h})\in\mathcal{A}(E_{h})$ as

[TABLE]

where $\sigma_{h}\to 1$ is such that $\int\widetilde{\rho}_{h}=1$. Since

[TABLE]

by testing with $u_{E}-w_{h}$ and by exploiting the Sobolev embedding we obtain

[TABLE]

Then Young’s inequality implies that $\|\nabla(u_{E}-w_{h})\|_{2}\to 0$. Since also $\|\widetilde{\rho}_{h}-\rho_{E}\|_{2}\to 0$ and $(w_{h},\widetilde{\rho}_{h})\in\mathcal{A}(E_{h})$, we get that

[TABLE]

Strong convergence of $\nabla u_{E_{h}}$ and $\rho_{E_{h}}$ is now a simple consequence of the convergence of energies. ∎

3. Small volume adjustments and Euler Lagrange equations

In this section we show how to adjust the volume of a given set without increasing too much its energy which will be instrumental both to prove compactness of the class of minimizers in Section 5 and to get rid of the volume constraint in studying regularity of solutions of ($\mathcal{P}_{\beta,K,Q,R}$), see Section 4. The “adjustment” lemma will be proved with the aid of a deformation via a family of diffeomorphism close to the identity. Though not needed in the sequel we also establishes the Euler Lagrange equations associated to ($\mathcal{P}_{\beta,K,Q,R}$). We start with the following lemma.

Lemma 3.1**.**

For every $\eta\in C_{c}^{\infty}(B_{R};\mathbb{R}^{n})$ there exists $t_{0}=t_{0}(\operatorname{dist}(\operatorname{spt}\eta,\partial B_{R})>0$ such that $\{\varphi_{t}\}_{|t|\leq t_{0}}$ defined by $\varphi_{t}(x):=x+t\,\eta(x)$ is a family of diffeomorphisms of $B_{R}$ into itself. Moreover for some set $E\subset B_{R}$ let $(u,\rho)$ be a solution of

[TABLE]

Then setting

[TABLE]

we have

[TABLE]

where $\|A_{t}-\operatorname{Id}\|_{\infty}=O(t)$ and the implicit constant depends only on $\|\nabla\eta\|_{\infty}$.

Proof.

The proof of the first part of the Lemma is straightforward. For the second we see that for $\psi\in C_{c}^{\infty}$, by change of variables $x=\varphi_{t}(y)$,

[TABLE]

Where we have used the equality $\nabla\varphi\circ\varphi_{t}^{-1}=(\nabla\varphi_{t}^{-1})^{-1}$ and for a matrix $N$ we denoted by $N^{T}$ its transpose and by $N^{-T}$ for $(N^{-1})^{T}$. Hence $u_{t}$ is a solution of 3.1 with

[TABLE]

By explicit computation we see that $A_{t}$ satisfies the desired bound. ∎

We now show how the energy $\mathcal{G}$ changes by the effect of a family of diffeomprohism.

Lemma 3.2**.**

Let $E\subseteq B_{M}$ be a measurable set and let $\{\varphi_{t}\}_{|t|\leq t_{0}}$ be a family of diffeomorphisms as in Lemma 3.1. Then

[TABLE]

where $E_{t}:=\varphi_{t}\left(E\right)$ and the implicit constant depends only on $\|\nabla\eta\|_{\infty}$. Moreover

[TABLE]

Proof.

Let $(u_{E},\rho_{E})\in{}\mathcal{A}(E)$ be a the optimal pair for $\mathcal{G}_{\beta,K}(E)$. By Lemma 3.1 $u_{t}=u_{E}\circ{}\varphi^{-1}_{t}$ solves (3.1) with $\widetilde{\rho}_{t}=\rho_{E}\circ\varphi_{t}^{-1}\det(\nabla\varphi_{t}^{-1})$. Let $v_{t}$ be the solution of

[TABLE]

Step 1: We start by proving the following estimate

[TABLE]

where the implicit constant depends only on $\|\nabla\eta\|_{\infty}$. In order to prove (3.5) we claim that

[TABLE]

Indeed assuming that (3.6) holds true and using that $|a|^{2}-|b|^{2}=2b\cdot{}(a-b)+|a-b|^{2}$ for every $a$,$b\in\mathbb{R}^{n}$, we have

[TABLE]

We estimate the first term in the right hand side of (3.7). By (3.6), we find that

[TABLE]

By (3.7) and (3.8), we have:

[TABLE]

which proves (3.5).

Let us now prove (3.6). By testing (3.1) and (3.4) with $v_{t}-u_{t}$ we get

[TABLE]

Rearranging terms and recalling that $|A_{t}-\operatorname{Id}|=O(t)$, this gives

[TABLE]

which, by Young’s inequality, implies (3.6).

Step 2: By changing of variables

[TABLE]

Moreover

[TABLE]

which gives

[TABLE]

and the more precise equality

[TABLE]

In the same way we get

[TABLE]

Furthermore, since $\det\nabla\varphi_{t}=1+t\operatorname{div}\eta+o(t)$, we also get

[TABLE]

Step 4: Since, by its definition

[TABLE]

and $v_{t}$ solves (3.4), we see that $(v_{t},\widetilde{\rho}_{t})\in\mathcal{A}(E_{t})$. Hence, by combining $\leavevmode\nobreak\ \eqref{stima1}$, $\leavevmode\nobreak\ \eqref{stima2}$ and $\leavevmode\nobreak\ \eqref{stima3}$ we obtain

[TABLE]

which proves (3.2). The proof of (3.3) is obtained by combining the above argument with (3.10) and (3.12). ∎

By combining the Taylor expansion of the perimeter, [19, Theorem 17.8],

[TABLE]

with (3.3) we obtain the Euler Lagrange equations for minimizers of ($\mathcal{P}_{\beta,K,Q,R}$) whose proof is left to the reader.

Corollary 3.3**.**

Let $E$ be a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), then

[TABLE]

for all $\eta\in C_{c}(B_{R};\mathbb{R}^{n})$ with $\int\operatorname{div}\eta=0$.

The next series of results are modeled after [2] and allow to do small volume adjustments without increasing too much the perimeter, see also [19, Chapter 17]. The first lemma is elementary.

Lemma 3.4**.**

Let $E\subseteq\mathbb{R}^{n}$ be a set of finite perimeter and let $U$ be an open set such that $P(E,U)>0$. Then there exists $\varepsilon=\varepsilon(E)>0$, $\gamma=\gamma(E)>0$ and a vector field $\eta_{E}\in C^{\infty}_{c}(U;\mathbb{R}^{n})$ with $\|\eta\|_{C^{1}}\leq 1$ such that

[TABLE]

Proof.

Since

[TABLE]

we find a vector filed such that

[TABLE]

Taking $\eta=\widetilde{\eta}/\|\widetilde{\eta}\|_{C^{1}}$ we obtain the desired conclusion. ∎

In order to have uniform controls on the constants involved in our regularity theory, we need to enforce the above lemma in the following one, based on a concentration compactness argument. Note that this time the constants depend only on the upper bound on the perimeter, in particular they do not depend on $R$.

Lemma 3.5**.**

For every $P>0$ there exist constants $\bar{\gamma}=\bar{\gamma}(n,P)>0$ and $\bar{\delta}=\bar{\delta}(n,P)$ such that if $R\in(1,\infty)$ and $E\subset B_{R}$ satisfies

[TABLE]

then there exists a vector filed $\eta\in C_{c}^{1}(B_{R})$ with $\|\eta\|_{C^{1}}\leq 1$ such that

[TABLE]

Proof.

Let us argue by contradiction: assume that there exist a sequence of radii $R_{k}$ and a sequence of sets $E_{k}$ satisfying (3.13) such that

[TABLE]

where $\bar{\delta}=\bar{\delta}(n,P)$ is a small constant to be fixed later only in dependence of $n$ and $P$. By [19, Remark 29.11] there exist points $y_{k}\in\mathbb{R}^{n}$ and a constant $\delta_{1}=\delta_{1}(n,P)$ such that

[TABLE]

Then by taking $z_{k}\in E_{k}\cap B_{1}(y_{k})\subset B_{R_{k}}\cap B_{1}(y_{k})$ we get

[TABLE]

Let us now detail the proof in the case in which, up to subsequences, $R_{k}\to\infty$ and $\partial B_{R_{k}-1}\cap B_{2}(z_{k})\neq\emptyset$. The other cases are actually simpler and we explain how to modify the argument at the end of the proof. We first note that since $\partial B_{R_{k}-1}\cap B_{2}(z_{k})\neq\emptyset$, we can take $x_{k}\in\partial B_{R_{k}}$ such that

[TABLE]

Now a simple geometric argument ensures that

[TABLE]

In particular we can chose $\delta_{2}=\delta_{2}(n,P)$ such that

[TABLE]

Let us now assume that, up to subsequences and a possible rotation of coordinates

[TABLE]

where the first limit exits due to our assumption on the perimeters. In particular

[TABLE]

and $F\subset\widehat{B}$. Note that by (3.15), $F\neq\emptyset$ and, since $|F_{k}|\leq 3|B_{1}|/2$, $|\widehat{B}\setminus F|>0$. In particular, $P(F,\widehat{B})>0$. By Lemma 3.4, we can find a constant $\gamma=\gamma_{F}>0$ and vector field $\eta_{F}\in C^{1}_{c}(\hat{B};R^{d})$ with $\|\eta\|_{C^{1}}\leq 1$ such that

[TABLE]

For $k$ large, the vector field $\eta_{k}=\eta_{F}(\cdot+x_{k})$ satisfies, $\eta_{k}\in C_{c}(B_{R_{k}-\delta_{2}/2})$, $\|\eta\|_{C^{1}}\leq 1$ and contradicts (3.14) with $\bar{\delta}=\delta_{2}/2$.

Let us conclude by explaining how to modify the proof in the case in which either $B_{2}(z_{k})\cap\partial B_{R_{k}-1}=\emptyset$ or $R_{k}\to\bar{R}<\infty$. In the first case instead one argue as above by considering the set $F_{k}=E_{k}\cap B_{2}(z_{k})-z_{k}$ and by noticing that the vector fields $\eta_{k}=\eta_{F}(\cdot+y_{k})$, $F=\lim F_{k}$, are compactly supported in $B_{R_{k}-1/2}$. In the second case one can simply reproduce the above argument. ∎

The next proposition will be crucial in removing the volume constraint and in making comparison estimates for minimizers of ($\mathcal{P}_{\beta,K,Q,R}$). The proof is based on a concentration-compactness argument.

Proposition 3.6**.**

For every $P>0$ there exist constants $\bar{\sigma}=\bar{\sigma}(n,P)>0$ and $C=C(n)$ such that if $R\in(1,\infty)$ and $E\subset B_{R}$ satisfies

[TABLE]

then for all $\sigma\in{}(-\bar{\sigma},\bar{\sigma})$ there exists $F_{\sigma}\subset B_{R}$ such that

[TABLE]

Proof.

By Lemma 3.5 we can find $\bar{\gamma}=\bar{\gamma}(n,P)>0$, $\bar{\delta}=\bar{\delta}(n,P)$ and a vector field \eta\in{}C^{1}_{c}\big{(}B_{R-\bar{\delta}};\mathbb{R}^{n}\big{)} with $\|\eta\|_{C^{1}}\leq 1$ such that

[TABLE]

Define a family of diffeomorphisms $\varphi_{t}:=\operatorname{Id}+t\,\eta$ and note that, since $\operatorname{dist}(\operatorname{spt}(\eta,\partial B_{R})\geq\bar{\delta}(n,P)$, they send $B_{R}$ into itself for $|t|\leq t_{0}(n,P)$ . By Taylor expansion

[TABLE]

and

[TABLE]

where the implicit constants depends only on $\|\nabla\eta\|_{\infty}\leq 1$. Moreover

[TABLE]

where $E_{t}=\varphi_{t}(E)$ and the constant in (3.18) depends only on $\|\nabla\eta\|_{\infty}\leq 1$. Hence we can find $t_{1}=t_{1}(n,P)>0$ such that

[TABLE]

for every $|t|\leq t_{1}$. By equations (3.19a) and (3.19b) we get

[TABLE]

Let $g(t):=|E_{t}|$ and note that thanks to (3.17) and (3.16), $g$ is increasing in a neighborhood of [math]. Take $\bar{\sigma}>0$ such that (|E|-\bar{\sigma},|E|+\bar{\sigma})\subseteq{}g\big{(}(-t_{1},t_{1})\big{)}. Then for every $|\sigma|\leq{}\bar{\sigma}$ there exists $t_{\sigma}>0$ such that $|E_{t_{\sigma}}|=|E|+\sigma$. Setting $F_{\sigma}=E_{t_{\sigma}}$ we obtain the desired conclusion. ∎

4. $\Lambda$-minimality and local variations

In order to study the regularity of minimizers it will be convenient to understand what is the behavior under small perturbations in balls. In this section we start by removing the volume constraint by showing that minizers are $\Lambda$-minimizer of $\mathcal{F}$ under small perturbations. In order to keep track of the dependence of the parameters in Theorem 1.2, it will be important that this “almost”-minimality depends only on the structural parameter of the problem. We start thus by fixing the following convention, which will be in force throughout all the rest of the paper:

Convention 4.1** (Universal constants).**

Given $A>0$, we say that $\beta,K,Q$ with $\beta\geq 1$ are controlled by $A$ if

[TABLE]

We will also say that a constant is universal if it depends only on the dimension $n$ and on $A$. Finally, for two positive quantities $X$ and $Y$, we will sometimes write $X\lesssim Y$ if there exists a universal constant $C$ such that $X\leq CY$ and we write $X\gtrsim Y$ if $Y\lesssim X$.

Note in particular that universal constants do not depend on the size of the container where the minimization problem is solved. Moreover we also remark here the following elementary fact: since $B_{1}$ is always a competitor for ($\mathcal{P}_{\beta,K,Q,R}$), if $E$ is a minimizer then

[TABLE]

whenever $\beta,K,Q$ are controlled by $A$.

Let us now introduce the following perturbed minimality condition.

Definition 4.2** ($(\Lambda,\bar{r})$-minimizer).**

We say that $E$ is a $(\Lambda,\bar{r})$-minimizer of the energy $\mathcal{F}$ if there exist constants $\Lambda>0$ and $\bar{r}>0$ such that for every ball $B_{r}(x)\subseteq\mathbb{R}^{n}$ with $r\leq\bar{r}$ we have

[TABLE]

Remark 4.3**.**

Note that if $E$ is $(\bar{\Lambda},\bar{r})$-minimizer than it is also a $(\bar{\Lambda}_{1},\bar{r}_{1})$-minimizer whenever $\bar{\Lambda}_{1}\geq\bar{\Lambda}$ and $\bar{r}_{1}\leq\bar{r}$. Hence there is no loss of generality in assuming that $\bar{r}\leq 1$.

We can now establish the desired $\Lambda$-minimality property for minimizers of ($\mathcal{P}_{\beta,K,Q,R}$).

Proposition 4.4**.**

Let $A>0$ and let $\beta,K,Q$ with $\beta\geq 1$ be controlled by $A$ and let $R\geq 1$. Then there exist $\Lambda_{1},\bar{r}_{1}>0$ universal such that all minimizers ($\mathcal{P}_{\beta,K,Q,R}$) satisfy

[TABLE]

whenever $F\subset B_{R}$ and $E\Delta F\subset B_{r}(x_{0})$, $r\leq\bar{r}_{1}$.

Proof.

Clearly we can suppose that

[TABLE]

since otherwise the result is trivial. In particular $P(F)$ is bounded by an universal constant $P$. Let $\bar{\sigma}$ and $C$ be the parameters in Proposition 3.6 associated to $P$. If $\bar{r}_{1}$ is chosen small enough we have

[TABLE]

Moreover, since $|E|=|B_{1}|$, $|F|\in(|B_{1}|/2,3|B_{1}|/2)$. Hence we can apply Proposition 3.6 to $F$ to obtain a set $\widetilde{F}\subset B_{R}$ such that $|\widetilde{F}|=|B_{1}|$ and

[TABLE]

where the first inequality is due to the minimality of $E$. Since $\mathcal{F}_{\beta,K,Q}(F)\lesssim 1$ and

[TABLE]

we obtain the conclusion for a suitable universal constant $\Lambda_{1}$.

∎

We conclude this section by establishing the following “local” minimality properties of minimizers ($\mathcal{P}_{\beta,K,Q,R}$). Note that in (ii) below we are not requiring $F$ to be contained in $B_{R}$.

Proposition 4.5**.**

Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. Then there exist universal constants $\Lambda_{2}$ and $\bar{r}_{2}$ such that all minimizers ($\mathcal{P}_{\beta,K,Q,R}$) satisfy the following two properties:

1. (i)

for every set of finite perimeter $F\subseteq{}E$ with $E\setminus{}F\subset{}B_{r}(x)$ and $r\leq{}\bar{r}_{2}$ it holds:

[TABLE] 2. (ii)

for every set of finite perimeter $F\supseteq{}E$ with $F\setminus{}E\subset{}B_{r}(x)$ and $r\leq{}\bar{r}_{2}^{\prime\prime}$ it holds:

[TABLE]

In particular,

[TABLE]

whenever $F\Delta{}E\subset{}B_{r}(x)$ with $r\leq{}\bar{r}_{2}$.

Proof.

We start proving (i). Let $E$ be a minimizer and $(u_{E},\rho_{E})$ be the minimizing pair for $\mathcal{G}(E)$. Let $F\subseteq{}E$ be such that $E\setminus{}F\subset{}B_{r}(x)$ with $r\leq{}\bar{r}_{1}$ where $\bar{r}_{1}$ is the constant defined in Proposition 4.4, by possibly choosing $r_{1}$ smaller, we can assume that

[TABLE]

Let us set

[TABLE]

and let $u$ be the solution of

[TABLE]

Note that $(u,\rho)\in\mathcal{A}(F)$ and thus, by using the $\Lambda$-minimality of $E$ established in Proposition 4.4,

[TABLE]

Item (ii) will then follow if we can prove

[TABLE]

and

[TABLE]

To prove (4.8) we estimate

[TABLE]

where in the first inequality we have used (4.7) and the definition of $\lambda_{F}$. By (2.4), $\|\rho_{E}\|_{\infty}\lesssim 1$ and this concludes the proof of (4.8).

Let us now prove (4.9). First note that

[TABLE]

Testing the equations satisfied by $u_{E}$ and $u$ with $u_{E}$ and $u$ respectively and subtracting the result we obtain also

[TABLE]

Subtracting (4.10) from two times (4.11) we get

[TABLE]

Moreover,

[TABLE]

Combining (4.12) and (4.13) we then obtain:

[TABLE]

We start to estimate the first term in the right hand side of (4.14). By using Proposition 2.3 and by arguing as in the proof of (4.8) the first term can be easily estimated as

[TABLE]

To estimate the second term in the right hand side of (4.14), we write

[TABLE]

Therefore

[TABLE]

By the Sobolev embedding and Young inequality (and recalling that $1\leq a_{F}\leq\beta$), the above inequality immediately imply

[TABLE]

By the definition of $\rho$, the second term is $\lesssim|E\setminus F|$ (note that $2/(2^{*})^{\prime}\geq 1$) while the first one is less than

[TABLE]

Since also the third term in (4.14) can be estimated by the above integral, this concludes the proof of (4.9).

Let us now prove (ii). Let $F\supseteq E$, note that $P(F\cap B_{R})\leq P(F)$ and that $(F\cap B_{R})\setminus E\subset F\setminus E$. Hence if we can prove (i) for subsets of $B_{R}$ we will get it for all sets. Let us then assume that $E\subseteq F\subseteq B_{R}$. By $\Lambda$-minimality of $E$

[TABLE]

Since, by Lemma 2.4, $\mathcal{G}_{\beta,K}(E)\geq\mathcal{G}_{\beta,K}(F)$ the conclusion follows. ∎

Remark 4.6**.**

We record here the following simple consequence of (4.6). Assume that $|\nabla u_{E}|^{2}\in L^{p}$, then (4.6) and Hölder inequality imply that for $F$ such that $F\Delta{}E\subset{}B_{r}(x)$ with $r\leq{}\bar{r}_{2}$,

[TABLE]

In particular if $p>n$, then $n-\frac{n}{p}>n-1$ and thus $E$ is a $\omega$ minimizers of the perimeter in the sense of [25]. Hence $\partial E$ is a $C^{1}$ manifold outside a singular closed set $\Sigma$ of dimension at most $(n-8)$. Note that by Cordes estimate, [4], the assumption $|\nabla u_{E}|^{2}\in L^{p}$ with $p>n$ is satisfied wherever $\beta-1\ll 1$. In particular, in this regime, Taylor cones singularities are excluded in $\mathbb{R}^{3}$.

5. Compactness of minimizers

In this section we prove that the class of minimizers of ($\mathcal{P}_{\beta,K,Q,R}$) is a compact subset of $L^{1}$, this is not really necessary in the proof of the main result, but we believe it can be interesting by its own.

Proposition 5.1**.**

Let $K_{h},Q_{h}\in\mathbb{R}$, $\beta_{h}\geq 1$ and $R_{h}\geq 1$ be such that

[TABLE]

For every $h\in\mathbb{N}$ let $E_{h}$ be a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$) (with $\beta,K,Q,R$ replaced by $\beta_{h},K_{h},Q_{h},R_{h}$, then, up to a non relabelled subsequence, there exists a set of finite perimeter $E$ such that

[TABLE]

Moreover $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$) and

[TABLE]

Proof.

Since if $R_{h}=1$ for all $h$ the problem is trivial (recall that $|E_{h}|=|B_{1}|$) we can assume that $R_{h}$ and $R$ are strictly bigger than one. Moreover $B_{1}$ is always an admissible competitor and thus,

[TABLE]

In particular the perimeters of $E_{h}$ are uniformly bounded and since all the sets are included in, say, $B_{2R}$ there exists a non relabelled subsequence and set $E\subset B_{R}$ such that (5.1) hold true. Since the perimeter is lower-semicontinuous and, by Proposition 2.6, $\mathcal{G}$ is continuous we also get that

[TABLE]

We now show that $E$ is a minimizer. For let $F\subset B_{R}$ with $|F|=|B_{1}|$. Since $R_{h}\to R$, we can find $\lambda_{h}\to 1$ such that $F_{h}:=\lambda_{h}F\subset B_{R_{h}}$. Clearly, $F_{h}\to F$, $|F_{h}|=|F|+o(1)$ and $P(F_{h})=P(F)+o(1)$. Thus

[TABLE]

By Proposition 3.6 applied to $F_{h}$ we can find sets $\widetilde{F}_{h}\subset B_{R_{h}}$ such that $|\widetilde{F}_{h}|=|B_{1}|$ and

[TABLE]

where in the last equality we have used (5.3). By minimality of $E_{h}$ we get

[TABLE]

which combined with (5.2) implies the minimality of $E$. By choosing $E=F$ we also deduce the convergence of the energies and, by Proposition  2.6, this implies the convergence of the perimeters. ∎

6. Decay of the Dirichlet energy and density estimates

6.1. Decay of the Dirichlet energy

Following [12], in this subsection we establish an almost Lipschitz decay for the Dirichlet energy of $u_{E}$ in certain regimes. Namely when the set or the complement almost fill a ball or when the set is very close to an half space.

We start by recalling the following higher integrability lemma for solution of (1.6) The proof can be found for instance in [14].

Lemma 6.1**.**

Let $E$ be set of finite measure and let $(u,\rho)\in\mathcal{A}(E)$. Then there exists $C=C(n,\beta)$ and $p=p(n,\beta)>1$ such that for all balls $B_{r}(x)\subset\mathbb{R}^{n}$

[TABLE]

Furthermore, the constants $C$ and $p$ depends only on an upper bound for $\beta$.

We start with the following elementary lemma where the optimal decay is obtained in some limit situations.

Lemma 6.2**.**

Let $\beta\geq 1$ and $\rho\in L^{\infty}(\mathbb{R}^{n})$. Then here exists a dimensional constant $C=C(n)$ such that:

1. (i)

if $v\in W^{1,2}(B_{r}(x))$ is a solution of

[TABLE]

then for all $\lambda\in(0,1)$

[TABLE] 2. (ii)

If $v\in W^{1,2}(B_{r}(x))$ is a solution of

[TABLE]

where H:=\bigl{\{}y\in\mathbb{R}^{n}:(y-x)\cdot e\leq 0\bigr{\}} for some $e\in\mathbb{S}^{n-1}$. Then for all $\lambda\in(0,1)$

[TABLE]

Proof.

We just prove point (ii) since (i) is a particular case (and well known). By scaling and translating, we can assume without loss of generality that $x=0$ and $r=1$. Let $w$ be the solution of

[TABLE]

so that $u=v-w$ solves

[TABLE]

By multiplying the last equation by $u$, applying Poincaré inequality we obtain

[TABLE]

where we have used that $a_{H}\geq 1$. Hence

[TABLE]

Moreover, by [12, Lemma 2.3],

[TABLE]

Hence,

[TABLE]

which together with (6.4) concludes the proof. ∎

As in [12], we now exploit the higher integrability of $\nabla u_{E}$ recalled in Lemma 6.1 to obtain an “almost version” of the above decay.

Proposition 6.3** (Decay of Dirichlet energy).**

Let $\beta\geq 1$ then there exists a constant $C=C(n,\beta)$ with the following property: if $E\subset\mathbb{R}^{n}$, $u$ and $\rho$ satisfy

[TABLE]

then for all $\lambda\in{}\left(0,\frac{1}{2}\right)$ there exists $\varepsilon_{0}=\varepsilon_{0}(\lambda,\beta)>0$ such that

1. (i)

if

[TABLE]

then

[TABLE] 2. (ii)

If

[TABLE]

where H:=\bigl{\{}y\in\mathbb{R}^{n}:(y-x)\cdot e\leq 0\bigr{\}} for some $e\in\mathbb{S}^{n-1}$, then

[TABLE]

Moreover the constants $C$ and $\varepsilon_{0}$ can be chosen to depend only on un upper bound on $\beta$.

Proof.

We detail the proof of item (ii). Item (i) can be obtained in a similar way and we sketch the argument at the end of the proof. Without loss of generality, by scaling and translating, we can assume $x=0$ and $r=1$. Let $\lambda\in(0,1/2)$ be given and let $v$ the solution of

[TABLE]

where $a_{H}=\beta\mathbf{1}_{H}+\mathbf{1}_{H^{c}}$. In particular, $w=(u-v)\in W^{1,2}_{0}(B_{1/2})$ and

[TABLE]

By testing the above equation with $w$ and using Young inequality we get

[TABLE]

Exploiting the higher integrability of Lemma 6.1 we then get

[TABLE]

Since the decay estimate (6.3) apply to $v$, we can argue as in the proof of 6.5 to obtain

[TABLE]

Choosing $\varepsilon_{0}=\varepsilon_{0}(n,\lambda)\ll\lambda$ sufficiently small we conclude the proof of (ii). The proof of (i) can be obtained in the same way by comparing $u$ to a solution of $-\Delta u=\rho$ (or $-\beta\Delta u=\rho$) and by using (6.2). ∎

6.2. Density estimates

In this section we establish scaling invariant upper and lower bounds for the perimeter and for the measure of a minimizer in balls. We also establish an universal upper bound for the normalized Dirchlet energy of the minimizer of $u_{E}$. We start with the following lemma which is a simple consequence of the outward minimizing property of $E$ established in Lemma 4.5 (ii).

Proposition 6.4**.**

*Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. Then there exist universal constants $C_{\mathrm{o}}$ and $r_{\mathrm{o}}$ such that, if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), $r\in(0,r_{\mathrm{o}})$, then333Here and in the sequel we will always work with the representative of $E$ such that

\partial E=\Biggl{\{}x:\frac{|B_{r}(x)\setminus E|}{|B_{r}(x)|}\cdot\frac{|B_{r}(x)\cap E|}{|B_{r}(x)|}>0\quad\text{for all r>0}\Biggr{\}},

see [19, Proposition 12.19].*

[TABLE]

and

[TABLE]

Proof.

We let $\Lambda_{2}$ and $\bar{r}_{2}$ be the constants appearing in Lemma 4.5 we take $r_{\textrm{o}}\leq\bar{r}_{2}$. For $r\leq r_{\textrm{o}}$, we plug $F=E\cup B_{r}(x)$ in (4.5) and we obtain, after simple manipulations,

[TABLE]

Hence, assuming that $\Lambda_{2}r_{\textrm{o}}\leq 1$, we immediately get $P(E,B_{r}(x))\lesssim r^{n-1}$. To obtain the lower density bound for $E^{c}$ we set $m(r):=|B_{r}(x)\setminus E|$ and we use the isoperimetric inequality to deduce

[TABLE]

where we have used that, by co-area formula $m^{\prime}(r)=\mathcal{H}^{n-1}(\partial B_{r}(x)\setminus E)$. If we choose $r_{\textrm{o}}$ such that $Cm(r)^{\frac{1}{n}}\leq C(n\omega_{n})^{\frac{1}{n}}r_{\textrm{o}}\leq 1/2$ where $C$ is the implied universal constant in the above estimate, we obtain

[TABLE]

Since $x\in\partial E$, $m(r)>0$ for all $r>0$ then the above inequality implies that

[TABLE]

Hence $m(r)\gtrsim r^{n}$ and this concludes the proof. ∎

The next lemma establish an universal bound on the normalized Dirichlet integral.

Lemma 6.5**.**

Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. Then there exists a universal constant $C_{\mathrm{e}}$ such that, if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), then for all $x\in\overline{B_{R}}$,

[TABLE]

Proof.

The estimates is clearly true if $r\geq r_{0}$ where $r_{0}=r_{0}(n,A)$ (recall that $Q^{2}\int|\nabla u_{E}|^{2}\leq\mathcal{F}_{\beta,K,Q}(E)\lesssim 1$). Hence we can assume that $r\leq r_{0}\ll 1$. We claim the following: there exist constants $\lambda=\lambda(n,A)\in(0,1/2)$, $C=C(n,A)$ and $r_{0}=r_{0}(n,A)$ such that

1. (a)

If $x\in\partial B_{R}$ and $r\leq r_{0}$, then

[TABLE] 2. (b)

If $x\in B_{R}$ and r\leq\min\bigl{\{}\operatorname{dist}(x,\partial B_{R}),r_{0}/2\bigr{\}}, then

[TABLE]

Let $\varepsilon\ll 1$ to be fixed and let $r_{0}=r_{0}(\varepsilon)\ll\bar{r}_{1}$ where $\bar{r}_{1}$ is the constant in Proposition 4.4 and such that the following holds true

[TABLE]

where $H_{x}:=\{y:(y-x)\cdot x\leq 0\}$ is the supporting half space of $B_{R}$ at $x$. Note that since the curvatures of $\partial B_{R}$ are universally bounded (recall that $R\geq 1$), this can be achieved by choosing $r_{0}$ small only in dependence of $\varepsilon$.

Let now $x\in\overline{B_{R}}$ and $r\leq r_{0}$ be a radius satisfying either condition (a) (if $x\in\partial B_{R}$) or condition (b) (if $x\in B_{R}$) above. Let $(u_{E},\rho_{E})$ be the minimizers for $\mathcal{G}(E)$ and consider

[TABLE]

We define $u$ to be the solution of

[TABLE]

Note that $(u,\rho_{E})\in\mathcal{A}(F)$ since $F\supset E$. Hence, by Proposition 4.4,

[TABLE]

where we have used that $F\setminus E\subset B_{r}(x)$ and that $P(F)\leq P(E\cup B_{r}(x))$, by the convexity of $B_{R}$. Rearranging terms we get

[TABLE]

Recall now that $u_{E}$ solves

[TABLE]

and we use (2.9) in Lemma 2.4 to infer that

[TABLE]

Since

[TABLE]

by testing with $u_{E}-u$ and by Young inequality we get

[TABLE]

where the last inequality follows from (6.13).

We want now apply Lemma 6.3 to $u$. Note that since

[TABLE]

then the assumption are satisfied both in case (a) (thanks to (6.11)) and in case (b) (since $B_{r}(x)\subset B_{R}$). Hence, given $\lambda\in(0,1/2)$, we have:

[TABLE]

for a constant $C=C(n,A)$ provided $\varepsilon$ (and thus $r_{0}$) is chosen sufficiently small. Since by (2.4) $\|\rho_{E}\|_{\infty}\lesssim 1$, we deduce from (6.14) that

[TABLE]

Now choosing $\lambda=\lambda(n,A)$ such that $C\lambda=1/2$ we conclude the proof of the claim. Note that this fixes $\varepsilon$ and thus $r_{0}$ as functions depending only on $n$ and $A$.

To conclude the proof we have to show that (a) and (b) above implies that

[TABLE]

We first assume that $S<+\infty$ and show that we can bound it by a universal constant. Let $\bar{y}\in\overline{B_{R}}$ and $\bar{s}\in 0<s\leq r_{0}$ be such that

[TABLE]

Let us distinguish a few cases:

$\bullet$ Case 1: $\bar{y}\in\partial B_{R}$. If $\bar{s}\leq\lambda r_{0}$, (6.9) implies that

[TABLE]

and we are done. On the other end if $\bar{s}\geq\lambda r_{0}$, then

[TABLE]

$\bullet$ Case 2: $\bar{y}\in B_{R}$. If $\bar{s}\leq\lambda\min\{\operatorname{dist}(\bar{y},\partial B_{R}),r_{0}/2\}$, we can use (6.10) and we argue as in the first part of Case 1. If $\bar{s}\geq\lambda r_{0}/2$ we argue instead as in the second part of Case 1 to conclude. We are thus left to consider the case

[TABLE]

In this case $B_{\bar{s}}(\bar{y})\subset B_{r_{0}}(\bar{y})$, $\bar{y}\in\partial B_{R}$ and

[TABLE]

Thus we are done.

To show that one can actually assume that $S<+\infty$ one can consider

[TABLE]

and argue as above to show that $S_{\delta}\leq C(n,A)$. Letting $\delta\to 0$ we conclude the proof. ∎

We are now ready to complete the proof of density and perimeter estimates.

Proposition 6.6**.**

Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. Then there exist universal constants $C_{\mathrm{i}}$ and $\bar{r}_{\mathrm{i}}$ such that, if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), then

[TABLE]

and

[TABLE]

Proof.

We start showing the validity of (6.16) and we divide the proof in few steps.

$\bullet$ Step 1: We claim that for every $\lambda\in(0,1/4)$, there exist $\varepsilon_{1}=\varepsilon_{1}(\lambda,,A)$, $C_{1}=C_{1}(n,A)$ and and $\bar{r}=\bar{r}(n,A,\lambda)$ such that if

[TABLE]

then,

[TABLE]

For the ease of notation let us assume that $x=0$. Let $\lambda\in(0,1/4)$ be fixed. By the relative isoperimetric inequality

[TABLE]

By (6.7) and by choosing $\varepsilon_{1},\bar{r}\ll 1$ we get

[TABLE]

Let us choose $t\in{}(\lambda{}r,2\lambda{}r)$ such that

[TABLE]

By testing (4.5) with $F=E\setminus{}B_{t}(x)$ we obtain

[TABLE]

which together with (6.20) and recalling that $t\in(\lambda r,2\lambda r)$, implies that

[TABLE]

If we now choose $\varepsilon_{1}=\varepsilon_{1}(\lambda)\ll 1$, (6.19) allow to apply Proposition 6.3 (i). Hence by also choosing $\bar{r}\ll\lambda$ we deduce that

[TABLE]

where we have used that by (2.4), $\|\rho_{E}\|_{\infty}\lesssim 1$. By gathering equations (6.22) and (6.23) we then get

[TABLE]

If we choose $\varepsilon_{1}=\varepsilon_{1}(n,A,\lambda)\ll 1$ such that $C(n,\lambda)\varepsilon^{\frac{1}{n-1}}\leq\lambda^{n}$ the above inequality implies (6.18).

$\bullet$ Step 2: We now prove the validity of (6.16). By density it is enough to prove it at all $x\in\partial^{*}E$. Again we set coordinates so that $x=0$. Let us choose $\lambda=\lambda(n,A)\in(0,1/4)$ such that $C_{1}\lambda\leq 1/2$ where $C_{1}$ is the constant appearing in (6.18) and let $\bar{r}$ and $\varepsilon_{1}$ be the corresponding constants (which now depend only on $A$ and $n$). We claim that

[TABLE]

Indeed otherwise, by (6.18) and the choice of $\lambda$

[TABLE]

We can thus iterate the above estimate and deduce that

[TABLE]

in contradiction with the assumption that $0\in\partial^{*}E$. Let now $\bar{\lambda}\ll\varepsilon_{1}$ to be chosen where $\varepsilon_{1}$ is the constant obtained above. Let $\varepsilon_{2}$ and $r_{2}$ be the constants corresponding to $\bar{\lambda}$ in Step 1. We claim that if we choose $\bar{\lambda}$ small enough depending only on $n$ and $A$ then

[TABLE]

where $r_{3}\ll\min\{r_{2},r_{1}\}$ will depend only on $n$ and $A$. Indeed otherwise we can apply Step 1, (6.6), and Lemma 6.5 to get

[TABLE]

where $\varepsilon_{2}\ll\varepsilon_{1}$ and $r_{2}\ll r_{1}$ are universal constants. If $\bar{\lambda}$ is chosen so that $\bar{C}(n,A)\bar{\lambda}\leq\varepsilon_{1}/4$ this contradicts (6.24) and thus proves (6.16) with $c_{\mathrm{i}}\leq\varepsilon_{2}$.

$\bullet$ Step 3: We now prove the validity of (6.17). Assume indeed that

[TABLE]

with $\varepsilon_{4},r_{4}\ll 1$ to be fixed only in term of $n$ and $A$. Then, by (6.1) and (6.8), for all $s\in(r/4,r/2)$

[TABLE]

Moreover, by co-area formula, there exists $s\in(r/4,r/2)$ such that

[TABLE]

By testing (4.4) with $E\setminus B_{s}$ we get

[TABLE]

which together with (6.26) and (6.27) and provided $r_{4}\ll\varepsilon_{4}\ll 1$ implies

[TABLE]

for a suitable universal constant $C$. Choosing $\varepsilon_{4}$ small with respect to $\varepsilon_{2}$ we get

[TABLE]

in contradiction with (6.25).

∎

7. Decay of the excess

In this section we prove Theorem 1.2. Since the seminal works of De Giorgi and Almgren, [5, 1] the proof is based on an excess decay theorem, namely

Theorem 7.1** (Excess improvement).**

Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. There exists a universal constant $C_{\mathrm{dec}}>0$ such that for all $\lambda\in(0,1/4)$ there exists $\varepsilon_{\mathrm{dec}}=\varepsilon_{\mathrm{dec}}(n,A,\lambda)>0$ satisfying the following: if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$) and

[TABLE]

then

[TABLE]

Where we recall the definition of spherical excess

[TABLE]

and of the normalized Dirichlet energy

[TABLE]

As it is customary, the proof of the above theorem is based on “harmonic approximation” technique. More precisely we will go through the following steps:

1. (i)

In the small excess regime, the boundary of $E$ can be well approximated by the graph of a Lipschitz function $f$ with Dirichlet energy bounded by the excess. 2. (ii)

If the excess and the normalized Dirichlet of $u_{E}$ are small, $f$ is almost harmonic. 3. (iii)

Almost harmonicity of $f$ implies closedness to an harmonic function $g$ in the $L^{2}$ topology. By classical estimates for harmonic functions, an $L^{2}$ type of excess of $g$ decays which in turn implies the decay of the flatness $\mathbf{f}$ of $E$, see (7.17) below for the definition. 4. (iv)

Via a Caccioppoli type inequality, the decay of the flatness can be transferred to the decay of the excess. 5. (v)

Via Proposition 6.3 (ii), the decay of the excess implies the decay of the normalized Dirchlet energy.

Usually, Step (i) is obtained by reproducing at most points and at all scale an height type bound for $\partial E$ in the small excess regime and it thus relies on the scaling invariance of the problem studied. Step (ii) and (iv) are obtained by simple comparison arguments and Step (iii) is based on a compactness argument together with the classical regularity theory for harmonic functions.

In our situation the problem does not enjoy of a nice scaling behaviour, due to the global constraint $\int\rho_{E}=1$. However, the local estimates obtained in the previous section are exactly what we need to carry on the proof of Step (i), see Lemma 7.2 below. Since beside this fact, most of the proofs of the needed lemmas are almost verbatim adaptation of those present in literature, we will not detail all of them and we will just focus on the key points and on the main differences.

7.1. Lipschitz approximation

In this subsection we prove the Lipschitz approximation lemma. Let us first fix a few notations that will be useful through all the section.

For $\nu\in{}\mathbb{S}^{n-1}$ we let $\mathbf{p}^{\nu}(x):=x-(x\cdot{}\nu)\,\nu$ and $\mathbf{q}^{\nu}(x):=(x\cdot{}\nu)\,\nu$ be, respectively, the orthogonal projection onto the plane $\nu^{\perp}$ and the projection on $\nu$. For simplicity we denote $\mathbf{p}(x):=\mathbf{p}^{e_{n}}(x)$ and $\mathbf{q}(x):=\mathbf{q}^{e_{n}}(x)=x_{n}$.

We define the cylinder with center at $x\in{}\mathbb{R}^{n}$ and radius $r>0$ with respect to the direction $\nu\in{}\mathbb{S}^{n-1}$ as

[TABLE]

We will write $\mathbf{C}_{r}:=\mathbf{C}(0,r,e_{n})$ and $\mathbf{C}:=\mathbf{C}_{1}$. We will also will denote the ($n-1$)-dimensional disk centered at $y$ and of radius $r$ by

[TABLE]

For simplicity we will write $\mathbf{D}_{r}:=\mathbf{D}(0,r)$ and $\mathbf{D}:=\mathbf{D}(0,1)$. We also recall the definition of cylindrical excess in a direction $\nu\in\mathbb{S}^{n-1}$ to be

[TABLE]

so that

[TABLE]

The following height bound is crucial in the sequel. Note that it does not require any minimality property on $E$, only the validity of inequality (7.3) at all scales.

Lemma 7.2**.**

Let $C>0$, there exists an increasing function $\omega_{C}:(0,1)\rightarrow{}\mathbb{R}$ with $\omega_{C}(0^{+})=0$ depending only on $C$ , such that every $E\subseteq{}\mathbb{R}^{n}$ of finite perimeter in $\mathbf{C}(x,2r)$ such that

• (i)

$x\in\partial E$,

• (ii)

for all $y\in\partial E$ and $s$ such that $B_{s}(y)\subset{}\mathbf{C}(x,2r)$

[TABLE]

satisfies the following

[TABLE]

Proof.

Note that the assumptions are scaling and translation invariant, hence we can assume that $x=0$ and $r=1$. For every $t\in{}(0,1)$ let

[TABLE]

For every $E\subseteq{}\mathbb{R}^{n}$ let us call

[TABLE]

Define the functions $\omega_{1},\omega_{2},\omega_{3}:(0,1)\rightarrow{}\mathbb{R}$ as

[TABLE]

Let $\omega_{C}:=\max\{\omega_{1},\omega_{2},\omega_{3}\}$. Notice that $\omega_{C}$ is increasing since it is the maximum of increasing functions and by definition it satisfies (7.4), (7.5), and (7.6). Let us prove that $\omega_{C}(0^{+})=0$. Assume by contradiction that $\lim_{t\to{}0^{+}}\omega_{C}(t)>0$ then there exist a there exists a sequence $t_{k}\searrow{}0$ and $L>0$ such that $\omega_{C}(t_{k})>L$ for all $k$. We now distinguish three cases.

Case 1: Up to subsequences $\omega(t_{k})=\omega_{1}(t_{k})$ for every $k\in\mathbb{N}$. For every $k$ there exists $E_{k}\in\mathcal{M}_{t_{k}}$ such that $h_{E_{k}}\geq{}L$. By  (7.3)) up to subsequences there exists a set of finite perimeter $E\subseteq{}\mathbb{R}^{n}$ such that $E_{k}\cap{}\mathbf{C}_{r}\rightarrow{}E\cap\mathbf{C}_{r}$ whenever $r<2$ and

[TABLE]

Now take $\mathbf{C}_{s}\subset{}\mathbf{C}_{r}\subset{}\mathbf{C}_{2}$ with $s>1$. By the lower semicontinuity of the excess we obtain that $\mathbf{e}(E,0,s,e_{n})=0$. Moreover let $\{x_{k}\}_{k\in{}\mathbb{N}}$ be a sequence such that $x_{k}\in{}\partial{}E_{k}\cap{}\mathbf{C}$ and let us assume that $x_{k}\to x$. By (ii) one easily deduce that

[TABLE]

which implies that $x\in\partial E$ (recall that we are working with the representative of $E$ such that $\partial E=\operatorname{spt}D\mathbf{1}_{E}$). This in particular implies that $0\in\partial E$. Since $\mathbf{e}(E,0,s,e_{n})=0$ we get that $H=\{x:\mathbf{q}x<0\}$. However, if $x_{k}\in\partial E_{k}$ is such that $|\mathbf{q}x_{k}|\geq{}L$, up to a subsequence, we can assume that $x_{k}\to\bar{x}\in\partial E=\{x:\mathbf{q}x=0\}$, a contradiction.

Case 2: Up to subsequence $\omega(t_{k})=\omega_{2}(t_{k})$ for every $k\in\mathbb{N}$. Hence for every $k$ there exists $E_{k}\in\mathcal{M}_{t_{k}}$ such that $g_{E_{k}}\geq{}L$. Note that if $\ell\in{}(0,L)$ then

[TABLE]

Hence  (7.10) implies that, up to extracting a subsequence,

[TABLE]

Indeed if by contradiction  (7.12) does not hold then for every $j\gg 1$ there exists $k_{j}\in\mathbb{N}$ such that $\mathbf{q}x\leq\frac{1}{j}$ for every $x\in{}\mathbf{C}\cap{}\partial E_{k_{j}}$. By (7.10), since $\{\mathbf{q}x>\frac{1}{j}\}$ is connected, then necessarily $\mathbf{C}\cap{}E_{k_{j}}\supseteq{}\{\mathbf{q}x>\frac{1}{j}\}.$ By letting $j\to{}+\infty$ we get (7.13).

By arguing as in Case 1, we have that $E_{k}\to\{\mathbf{q}x\leq 0\}$, hence (7.13) cannot hold. Hence (7.12) holds, which is again in contradiction with Case 1.

Case 3: Up to subsequence $\omega(t_{k})=\omega_{3}(t_{k})$ for every $k\in\mathbb{N}$. This case can be ruled out by arguing as in Case 2 (or by working with $E^{c}$ which satisfies the same assumption of $E$). Therefore $\omega$ is the required function. ∎

Once the “qualitative” height bound has been established, one can repeat verbatim the proof of the Lipschitz approximation in [19, Theorem 2.37] to deduce that in the small excess regime $\partial E$ is mostly covered by the graph of a Lipschitz function. Note that in the cited reference one has an explicit formula for $\omega$ (namely $\omega(t)\lesssim t^{1/(n-1)}$) however this plays at all no role in the proof, see also [7, Lemma 4.3].

Lemma 7.3** (Lipschitz approximation I).**

Fix $C>0$. Then there exists $\varepsilon_{\mathrm{L}}=\varepsilon_{\mathrm{L}}(n,C)>0$ and $C_{\mathrm{L}}=C_{\mathrm{L}}(n,C)>0$ with the following property: let $E$ be a set of finite perimeter in $\mathbf{C}(x,4r)$ satisfying $x\in\partial E$,

[TABLE]

and

[TABLE]

Then there exists a function $f:\mathbb{R}^{n-1}\to{}\mathbb{R}$ with

[TABLE]

such that, defining $\Gamma_{f}:=x+\{(z,f(z))\,:\,z\in{}\mathbf{D}_{r}\}$,

[TABLE]

*here $\omega_{C}$ is the function in Lemma 7.2. *

Note that if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), the assumption of the Lipschitz approximation lemma are satisfied with some universal constant $C$ by (6.6) and (6.16). Hence we can cover most of its boundary by the graph of a Lipschitz function $f$. Moreover a simple comparison argument implies that the laplacian of $f$ is small in a suitable negative norm. More precisely we have the following:

Proposition 7.4** (Lipschitz approximation II).**

Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. Then there exists universal constants $\varepsilon_{\mathrm{lip}}$, and $C_{\mathrm{lip}}$ and a “universal” increasing function (i.e. depending only on $n$ and $A$) $\omega_{\mathrm{lip}}$ with $\omega_{\mathrm{lip}}(0+)=0$ such that if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), $x\in\partial E$ and

[TABLE]

then there exists a function $f$ satisfying (7.14) and (7.15) with $C_{L}$ and $\omega_{C}$ replaced by $C_{\mathrm{lip}}$ and $\omega_{\mathrm{lip}}$ respectively. Moreover

[TABLE]

for every $\varphi\in C^{1}_{c}(\mathbf{D}_{r})$.

Proof.

Upper and lower perimeter estimates established in (6.6) and (6.16) ensure that in every cylinder $C(x,4r)$ centered at $x\in\partial E$, $E$ satisfies the assumption of Lemma 7.3 with a universal constant $C=C(n,A)$ provided $r$ is smaller than an universal radius $\bar{r}$. This proves the first part of the proposition. The second part follows by plugging in (4.6) $F:=\psi_{t}(E)$, $\psi_{t}(x)=x+t\varphi(\mathbf{p}x)e_{n}$, and by performing the same computations done in [19, Proof of Theorem 23.7]. ∎

7.2. The Caccioppoli inequality

By (7.16) one will deduce that under the assumption of Theorem 7.1, there exists an harmonic function $h:D_{r}\to\mathbb{R}$ which is close to $f$ in $L^{2}$. This closeness, together with the regularity theory for harmonic function will allow to deduce the decay of an $L^{2}$ type excess of $f$ and thus for $E$. In order to pass from the $L^{2}$ excess to the classical one, one needs to esablish a Caccioppoli type inequality. To this end, given a set $E$ and a vector $\nu\in S^{n-1}$ we define the flatness of $E$ at the point $x\in\mathbb{R}^{n}$, at the scale $r>0$ with respect to the direction $\nu\in\mathbb{S}^{n-1}$ as

[TABLE]

Proposition 7.5** (Caccioppoli inequality).**

Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. Then there exists universal constants $\varepsilon_{\mathrm{cac}}$, and $C_{\mathrm{cac}}$ such that if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), $x\in\partial E$, and

[TABLE]

then

[TABLE]

Proof.

The proof can be obtained by verbatim repeating the arguments of [19, Chapter 24] and using (4.6) instead of the perimeter minimality in the comparison estimate of [19, Equation 24.48].

∎

7.3. Dirichlet improvement

We now show that in the small excess regime there is fixed scale decay of the Dirichlet energy.

Proposition 7.6** (Decay of the Dirichlet energy).**

Let $A>0$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. There exists a universal constant $C_{\mathrm{d}ir}>0$ such that for all $\lambda\in(0,1/2)$ there exists $\varepsilon_{\mathrm{dir}}=\varepsilon_{\mathrm{dir}}(n,A,\lambda)$ satisfying the following: if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), $x\in\partial E$ and

[TABLE]

then

[TABLE]

Proof.

By (6.6) and (6.16) we have that if $r$ is universally small we can apply Lemma 7.2 to $E$ in $\mathbf{C}(x,r)$ to obtain a universal modulus of continuity $\omega$ such that for $H=\{y:\mathbf{q}(y-x)\leq 0\}$,

[TABLE]

By Lemma 6.3 (ii) (applied in $B_{r/2}(x)$) and the above inequality, for all $\lambda\in(0,1/2)$ we can choose $\varepsilon_{\mathrm{dir}}=\varepsilon_{\mathrm{dir}}(n,A,\lambda)$ sufficiently small such that

[TABLE]

where in the first inequality we have also exploited (2.4) and in the second the obvious inequality $D_{E}(x,r/2)\leq 2^{n-1}D_{E}(x,r)$. This concludes the proof. ∎

7.4. Excess improvement

In this section we prove Theorem 7.1.

Proof of Theorem 7.1.

We claim that there exists a universal constant $C_{\textrm{exc}}$ such that for all $\lambda\in(0,1/8)$ there exists $\varepsilon_{\textrm{exc}}=\varepsilon_{\textrm{exc}}(n,A,\lambda)$ satisfying the following: for all minimizers of ($\mathcal{P}_{\beta,K,Q,R}$) with $\beta,K,Q$ controlled by $A$ and $R\geq 1$ if $x\in\partial E$ the following holds

[TABLE]

Note that the above claim, combined with Proposition (7.6) immediately implies the conclusion of the Theorem. Let us assume hence that there exists $\lambda\in(0,1/8)$ a sequence of minimizers $E_{k}\subset B_{R_{k}}$ with parameters $\beta_{k},K_{k},Q_{k}$ controlled by $A$, radii $r_{k}$ and points $x_{k}\in\partial E_{k}$ such that

[TABLE]

but

[TABLE]

for a suitable universal constant $C_{\mathrm{exc}}$. Note that up to translating and rotating we can assume that $x_{k}=0$ and that

[TABLE]

We apply Proposition 7.4 to each $E_{k}$. Hence, there exists a sequence of $1$-Lipschitz functions $f_{k}:\mathbb{R}^{n-1}\to{}\mathbb{R}$ such that

[TABLE]

Let us set

[TABLE]

By the Poincaré -Wirtinger inequality and (7.21b),

[TABLE]

Hence there exists $g$ in $W^{1,2}(\mathbf{D}_{\frac{1}{2}})$ such that $g_{k}\rightharpoonup g$ weakly in $W^{1,2}(\mathbf{D}_{\frac{1}{2}})$ to some $g$ and strongly in $L^{2}(\mathbf{D}_{\frac{1}{2}})$. Moreover by (7.21d), for all $\varphi\in C_{c}^{1}(\mathbf{D}_{\frac{1}{2}})$

[TABLE]

where $\varphi_{r_{k}}(z)=r_{k}\varphi(z/r_{k})\in C_{c}^{1}(\mathbf{D}_{\frac{r_{k}}{2}})$ satisfies $\|\nabla\varphi_{r_{k}}\|_{\infty}=\|\nabla\varphi\|_{\infty}$. Hence $g$ is harmonic. By the mean value property and (7.22)

[TABLE]

By Taylor expansion,

[TABLE]

If $2\lambda\in(0,1/4)$ we can integrate the above inequality to get

[TABLE]

Recall that, by the mean value property of harmonic functions, for every $r\leq{}\frac{1}{2}$ we have

[TABLE]

Hence,

[TABLE]

which, by the definition of $g_{k}$ and changing variables implies

[TABLE]

Let us define

[TABLE]

and note that, by (7.21b), Jensen inequality and (7.21c)

[TABLE]

Since the $f_{k}$’s are $1$-Lipschitz, (7.25) implies

[TABLE]

and thus

[TABLE]

On the other hand, (7.21a), Lemma (7.2) and (7.26) imply

[TABLE]

Combining (7.27) and (7.28) we deduce that

[TABLE]

On the other hand, by the perimeter density estimates (6.6) and (7.26)

[TABLE]

Hence we can apply Proposition 7.5 in $B_{4\lambda r_{k}}$ to get that

[TABLE]

where in the first inequality we have used (7.2). Furthermore, by Proposition (7.6) applied in $B_{r_{k}}$ we have

[TABLE]

Combining (7.29), (7.30) and (7.31) we thus infer that

[TABLE]

in contradiction with (7.20) if $C_{\mathrm{exc}}$ is chosen big enough depending only on $n$ and $A$. ∎

8. Proof of Theorems 1.1 and 1.2

In this section we prove our main theorems, Theorem 1.2 is an immediate consequcne of the following slightly more general theorem.

Theorem 8.1**.**

Let $A>0$ $\vartheta\in(0,1)$, and let $\beta,K,Q$ be controlled by $A$ and $R\geq 1$. There exist constants $C_{\mathrm{reg}}(n,A,\theta)>0$ and $\varepsilon_{\mathrm{reg}}=\varepsilon_{\mathrm{reg}}(n,A,\theta)>0$ if $E$ is a minimizer of ($\mathcal{P}_{\beta,K,Q,R}$), $x\in\partial E$ $r>0$ and $\nu\in S^{n-1}$ are such that

[TABLE]

*then there exists a $C^{1,\vartheta}$ function $f:\mathbb{R}^{n-1}\to\mathbb{R}$ with 444 Here

$[\nabla f]_{\vartheta/2}:=\sup_{x\neq y}\frac{|\nabla f(x)-\nabla f(y)|}{|x-y|^{\frac{\vartheta}{2}}}.$

[TABLE]

such that

[TABLE]

Proof.

Given $\vartheta\in(0,1)$ we fix $\bar{\lambda}\in(0,1/8)$ be such that

[TABLE]

and we let $\bar{\varepsilon}$ be the corresponding $\varepsilon_{\mathrm{dec}}$ in Theorem 7.1. Note that $\bar{\varepsilon}$ depends only on $n$, $A$ and $\vartheta$. We now choose $\varepsilon_{\mathrm{reg}}$ so that for all $y\in\partial E\cap B_{r}(x)$

[TABLE]

Hence we can apply Theorem 7.1 and (8.1) to deduce that for all $y\in\partial E\cap B{r/2}(x)$,

[TABLE]

Iterating we get

[TABLE]

which implies

[TABLE]

By classical arguments this together with the density estimates (6.6) and (6.16), implies that for all $y\in B_{r}(x)\cap\partial E$ there exists $\nu_{y}$ such that

[TABLE]

and

[TABLE]

The last two display yield the desired conclusion, see for instance [19, Theorem 26.3] or [13, Theorem 4.8]. ∎

We can now prove Theorem 1.1 by following the arguments [12]

Proof of Theorem 1.1.

By Theorem 1.2, if we set

[TABLE]

then $\partial E\setminus\Sigma_{E}$ is a $C^{1,\vartheta}$ manifold for all $\vartheta\in(0,1/2)$. Hence we will conclude the proof if we show that

[TABLE]

for some $\eta=\eta(n,B)>0$. Recall that by Lemma 6.1, $|\nabla u_{E}|^{2p}\in L_{\mathrm{loc}}^{1}$ for some $p=p(n,B)>1$, hence, by Hölder inequality

[TABLE]

Hence, by [11, Theorem 2.10], $\mathcal{H}^{n-p}(\Sigma^{1}_{E})=0$. We now show that

[TABLE]

for all $\alpha>n-8$ which clearly concludes the proof. Let us fix $\alpha>n-8$ and assume the contrary. By [13, Proposition 11.3], there will be a point

[TABLE]

and a sequence $r_{k}\to 0$ such that

[TABLE]

where $\mathcal{H}_{\infty}^{s}$ is the infinity Hausdorff pre-measure. Let us set $E_{k}=(E-x)/r_{k}$ and note that by (6.6) and the above equation

[TABLE]

for all $s>0$ and

[TABLE]

where $\Sigma^{2}_{E_{k}}=(\Sigma_{E}^{2}-x)/r_{k}$. Up to subsequences, $E_{k}\to F$. We claim that $F$ is a local minimizer of the perimeter. Indeed if $G\Delta F\Subset B_{s}$, by averaging we choose $t\in(s,2s)$ such that

[TABLE]

With this choice, defining $G_{k}=(x+r_{k}G)\cap B_{r_{k}t}(x)\cup(E\setminus B_{tr_{k}}(x))$ and note that $E\Delta G_{k}\Subset B_{2sr_{k}}(x)$. Hence by (4.6) and classical computations

[TABLE]

which implies the desired minimality property. Moreover, by using $G=F$ we also deduce that $P(E_{k},B_{s})\to P(F,B_{s})$ for almost all $s>0$.

Let now $\Sigma_{F}$ be the singular set of $F$, and recall that, by the regularity theory for set of minimal perimeter [19, Part III], $\mathcal{H}^{\alpha}(\Sigma_{F})=\mathcal{H}_{\infty}^{\alpha}(\Sigma_{F})=0$. Hence by the definition of Hausdorff measure, for all $\delta>0$ there exists an open set $U_{\delta}$ such that

[TABLE]

We claim that there exists $k=k_{\delta}>0$ such that $\Sigma^{2}_{E_{k}}\cap B_{1}\subset U_{\delta}$ which will be in contradiction with (8.2) if $\delta$ is chosen small enough. Assume the claim is false, hence there is a sequence of points $\Sigma^{2}_{E_{k}}\cap B_{1}\ni y_{k}\to\bar{y}\in\overline{B_{1}}$ with $\operatorname{dist}(\bar{y},\Sigma_{F})>0$. It is easy to see that, by the lower perimeter estimates (6.16), $\bar{y}\in\partial F$. Hence by regularity, for all $\varepsilon>0$ there exists $r>0$ such that

[TABLE]

By perimeter convergence, this implies that, for $k$ large

[TABLE]

Choosing $\varepsilon\ll 1$ we can apply Theorem 1.2 to deduce that $x+r_{k}y_{k}\notin\Sigma_{E}^{2}$, i.e. $y_{k}\notin\Sigma^{2}_{E_{k}}$. This final contradiction concludes the proof.

∎