Uniform density estimates and $\Gamma$-convergence for the Alt-Phillips functional of negative powers

1.   Introduction

Energy functionals involving the Dirichlet integral of a density $u$ and a potential term $W(u)$

appear in various models in the calculus of variations. A classical example is the Allen-Cahn ^[1] energy given by the double-well potential

which is relevant in the theory of phase-transitions and minimal surfaces. In their celebrated result, Modica and Mortola ^[13] showed that $0$-homogenous rescalings of bounded minimizers $|u| \le 1$, converge up to subsequences to a $\pm 1$ configuration separated by a minimal surface, i.e.,

with $E$ a set of minimal perimeter. At the level of the energy, this result is expressed in terms of the Gamma-convergence of the rescaled energies

to a multiple of the perimeter functional $c_0 Per_{\Omega}(E)$.

Other examples of energies appear in the theory of free boundary problems. When the potential $W(t)$ is not of class $C^{1, 1}$ near a minimum point, say $t = 0$, minimizers can develop patches where they take this value. The boundary of such a patch $\partial\{u = 0\}$ is the free boundary. Two particular potentials of interest are given by

which corresponds to the obstacle problem (for a comprehensive survey see ^[14]), and by

which corresponds to the Bernoulli free boundary problem (see for example ^[2,3,8]). These can be viewed as part of the family of power-potentials

which were considered by Alt and Phillips ^[4] in the early 80's.

Recently in ^[10], we investigated properties of non-negative minimizers and their free boundaries for Alt-Phillips potentials of negative powers

These potentials are relevant in the applications, for example in liquid models with large cohesive internal forces in regions of low density. The upper bound $\gamma < 2$ is necessary for the finiteness of the energy.

In ^[10] we showed that minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials

have optimal $C^\alpha$ Hölder continuity. The free boundary

is characterized by an expansion of the type

where $d$ denotes the distance to $F(u)$ and $c_\alpha d^\alpha$ represents the explicit 1D homogenous solution. Furthermore, we showed that $F(u)$ is a hypersurface of class $C^{1, \beta}$ up to a closed singular set of dimension at most $n- k(\gamma)$, where $k(\gamma) \ge 3$ is the first dimension in which a nontrivial $\alpha$-homogenous minimizer exists. We also established the Gamma-convergence of a suitable multiple of the $\mathcal E_\gamma$ to the perimeter of the positivity set $Per_\Omega(\{u > 0\})$ as $\gamma \to 2$.

In this work we investigate in more detail the properties of minimizers as the parameter $\gamma$ tends to the critical value $2$, and make precise the connection between their free boundaries and the theory of minimal surfaces. In particular we establish density estimates and the uniform convergence (up to subsequences) of the free boundaries $F(u_k)$ to a minimal surface, for a sequence of bounded minimizers $u_k$ corresponding to parameters $\gamma_k \to 2$, see Corollary 4.2. Uniform convergence results in different settings were obtained by Caffarelli and Cordoba ^[7] for the Allen-Cahn energy and the convergence in (1.1), and by Caffarelli and Valdinoci ^[9] for the $s$-nonlocal minimal surfaces with $s \to 1$. We also refer the reader to other related works in similar contexts ^{[5,11,15,16,17]}.

The constants in the Hölder and density estimates obtained in ^[10] degenerate as $\gamma \to 2$. However, here we develop uniform estimates in $\gamma$, and for this it is convenient to rescale the potential term in the functional $\mathcal E_\gamma$ in a suitable way (see (2.1)). We further establish the Gamma-convergence to the Dirichlet-perimeter functional

which was studied by Athanasopoulous, Caffarelli, Kenig, Salsa in ^[6]. Heuristically, this shows that the cohesive term $W$ has the effect of surface tension as $\gamma \to 2$.

2.   Main results

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with Lipschitz boundary. We consider $J_\gamma$, a rescaling of $\mathcal E_\gamma$, which acts on functions

and it is defined as

where

We study uniform properties of the minimizers of $J_\gamma$ as $\gamma \to 2^-$. We often drop the dependence on $\gamma$ from $J$ and $W$ when there is no possibility of confusion.

Notice that $u$ is a minimizer of $\mathcal E_\gamma$ defined in (1.2), if and only if $c(\gamma) u$ is a minimizer of $J_\gamma$, with $c(\gamma) = c_\gamma^\frac{1}{\gamma+2}$ an appropriate constant depending only on $\gamma$, and $c(\gamma) \to 0$ as $\gamma \to 2$.

The constant $c_\gamma$ in (2.2) is chosen such that

The homogenous 1D solution $\varphi$ plays an important role in the analysis. It is given by

with

and satisfies

We differentiate the last equality and obtain that $\varphi$ solves the Euler-Lagrange equation

Positive constants depending only on the dimension $n$ are denoted by $c$, $C$, and referred to as universal constants.

The first result is an optimal uniform growth estimate.

Theorem 2.1. Let $u$ be a minimizer of $J_\gamma$ in $B_1$ and assume $u(0) = 0.$ Then, there exists a universal constant $C$ such that

The second theorem gives the uniform density estimate of the free boundary.

Theorem 2.2 (Density estimates). There exists a universal constant $c_0$ such that if $u$ is a nonnegative minimizer of $J_\gamma$ in $B_1$ and $0 \in F(u)$ then

The following result is a direct consequence of Theorems 2.1 and 2.2.

Corollary 2.3. Let $u$ be a nonnegative minimizer of $J_\gamma$ in $B_1$. If $0 \in F(u)$ then for all $r \in (0, 1/2)$ each of the sets $\{u = 0\} \cap B_r$ and $\{u > 0\} \cap B_r$ contains an interior ball of radius $c r$. Moreover

Next we introduce the Dirichlet-perimeter functional $\mathcal F$ introduced by Athanasopoulous, Caffarelli, Kenig, Salsa in ^[6]. It acts on the space of admissible pairs $(u, E)$ consisting of functions $u \ge 0$ and measurable sets $E \subset \Omega$ which have the property that $u = 0$ a.e. on $E$,

The functional $\mathcal F$ is given by the Dirichlet-perimeter energy

where $P_{\Omega}(E)$ represents the perimeter of $E$ in $\Omega$

The next theorem establishes the $\Gamma$-convergence of the $J_\gamma$'s.

Theorem 2.4. As $\gamma \to 2$, the functionals $J_\gamma$ $\Gamma$-converge to $\mathcal F$.

More precisely we have:

a) (lower semicontinuity) if $\gamma_k \to 2$ and $u_k$ satisfy

then

b) (approximation) given $(u, E) \in \mathcal A(\Omega)$ with $u$ a continuous in $\overline \Omega$, there exists $\gamma_k \to 2$ and $u_k$ such that

Our main result gives the strong convergence of the minimizers of $J_\gamma$ and their zero set to the minimizing pairs $(u, E)$ of $\mathcal F$.

Theorem 2.5. Let $\Omega$ be a bounded domain with Lipschitz boundary, $\gamma_k \to 2^-$, and $u_k$ a sequence of functions with uniform bounded energies

for some $M > 0$. Then, after passing to a subsequence, we can find $(u, E) \in \mathcal A(\Omega)$ such that

and

Moreover, if $u_k$ are minimizers of $J_{\gamma_k}$ then the limit $(u, E)$ is a minimizer of $\mathcal F$. The convergence of $u_k$ to $u$ and respectively of the free boundaries $\partial \{u_k > 0\}$ to $\partial E$ is uniform on compact sets (in the Hausdorff distance sense).

As a consequence we obtain the connection between bounded minimizers of $\mathcal E_\gamma$ with $\gamma \to 2$ and minimal surfaces, as stated in the Introduction. The uniform boundedness of minimizers can be deduced for example from a uniform bound of the boundary data on $\partial \Omega$.

Corollary 2.6. Assume that $u_k$ are uniformly bounded minimizers of $\mathcal E_{\gamma_k}$ defined in (1.2), and $\gamma_k \to 2$. Then, up to subsequences, $F(u_k)$ converge uniformly on compact sets to a minimal surface $\partial E$.

Indeed, $c(\gamma_k) u_k$ is a minimizer for $J_{\gamma_k}$ and, since $c(\gamma_k) \to 0$, the limiting function $u$ of Theorem 2.5 is identically $0$. This means that the limiting set $E$ must be a set of minimal perimeter in $\Omega$.

The paper is organized as follows. In Section 3 we prove the uniform growth estimate Theorem 2.1 and in Section 4 we obtain the uniform density estimates. In the last section we prove the main result Theorem 2.5.

3.   Proof of Theorem 2.1

In this section we prove Theorem 2.1. We state it here again for the reader convenience. We remark that this statement was proved in ^[10] with a constant $C$ depending on $\gamma$. The purpose of this section is to show that in fact the statement holds with a universal constant $C$. In the proof, we use that minimizers are viscosity solution in the sense of Definition 4.1 of ^[10], as showed in Proposition 4.4 of ^[10].

Theorem 3.1. Let $u$ be a minimizer of $J_\gamma$ in $B_1$, and assume $u(0) = 0.$ Then

with $C$ universal.

Proof. Minimizers of $J$ are invariant under $\alpha$-homogenous rescalings

After such a rescaling, we may assume that we are in the situation $B_1\subset \{u > 0\}$ and $u$ vanishes at some point $x_0 \in \partial B_1$. We need to prove that $u(0)$ is bounded above by a large universal constant.

Notice that in $B_1$ we satisfy

Thus, if

then by the weak Harnack inequality we find

Lemma 3.2. There exists a one dimensional increasing function $\psi$,

such that (see (2.4) for the definition of $\varphi$)

1)

2)

3)

Using Lemma 3.2 we construct a barrier $\Psi : B_1 \setminus B_{1/2} \to \mathbb R$, as

and

with boundary conditions

Since $\psi(t_0) \le 1$, it follows that

provided that $M$ is large universal.

We claim that

The inequality is satisfied in the outer annulus $B_1 \setminus B_{1-t_0}$ by property 2) above, and in the inner annulus $B_{ 1-t_0} \setminus B_{1/2}$ since $0 > W'$.

Moreover, the inequalities between the normal derivatives on either side of $\partial B_{ 1-t_0}$ guarantee that (3.1) holds in the whole domain.

Since $W'(t)$ is increasing for $t > 0$, we can apply the maximum principle and conclude that

We contradict the free boundary condition at the point $x_0 \in F(u)$ for a minimizer, see Proposition 4.4 in ^[10]. Indeed, property 1) above shows that $\Psi-\varphi(d)$ has a positive correction term $\epsilon \, d^{2-\alpha}$ in the expansion near its free boundary and therefore it is a strict viscosity subsolution on $\partial B_1$, see Definition 4.1 in ^[10]. □

It remains to prove the lemma above.

Proof of Lemma 3.2. We reduce the second order ODE to a 1st order ODE by taking $\psi$ as an independent variable. More precisely, with a strictly increasing function $\psi$ we associate the function $g > 0$ defined on the range of $\psi$ as

After differentiation we obtain

The function $\psi$ can be recovered from $g$ by the formula

In the case when $\psi$ coincides with the 1D solution $\varphi$ given in (2.4), then the associated function $g$ equals $W$, see (2.5).

In our setting we define $g$ explicitly as

with $C_1 = 8 \, n$ universal, and $s_0$ given by the solution to

and $\bar \epsilon > 0$ arbitrarily small. Notice that $s_0 \to 0$ as $\gamma \to 2^-$, and from the formula for $c_\gamma$ in (2.3) it follows

Notice that

This gives property 3) since

By construction $g \ge W$ which by (3.3) implies $\psi \ge \varphi$. Thus

and by (2.4), (3.4), it follows that also $t_0 \to 0$ as $\gamma \to 2^-$.

We compute

and use the inequality

and that $g \le 3 W$ in the interval $[0, s_0]$ to obtain

which gives 2).

Finally, we obtain property 1) from (3.3) and the expansion

□

4.   Density estimates

In this technical section we prove Theorem 2.2 and Corollary 2.3. We follow the classical ideas from the minimal surface theory by constructing appropriate competitors for the minimizer $u$, and then make use of the isoperimetric inequality. They allows us to obtain discrete differential inequalities for the measure of the sets $\{u > 0\}$ in $B_r$, which give the desired conclusion after iteration.

We start with the lower bound.

Lemma 4.1. Let $u$ be a minimizer of $J_\gamma$ in $B_1$ and assume $0\in F(u).$ Then

Proof. After a dilation, assume $u$ minimizes $J$ in $B_3$. Since $0 \in F(u)$,

by Theorem 2.1. Define

It suffices to show that

which is not possible since $0 \in F(u)$. We consider the case when $\gamma$ is close to 2.

Define $s_0$, $t_0$, as

and notice that $s_0, t_0 \to 0$ as $\gamma \to 2^-$.

Step 1: We show that the densities of the sets $A_r$ in $B_r$ decay geometrically as we rescale by a factor of $1 - 2 t_0$, i.e., if $a(1) \le c_0$ then

First we construct a 1D function.

Lemma 4.2. There exists a piecewise $C^1$ function $\psi$ in $[0, 1]$ such that

1)

2)

3)

Recall that $\psi$ being piecewise $C^1$ means that it is continuous in $[0, 1]$ and $C^1$ when restricted to the intervals $[0, t_0]$ and $[t_0, 1]$.

Proof. Indeed, we may take

with $g$ an increasing $C^2$ function in $[t_0, 1]$ such that

and $K$ a sufficiently large universal constant. Properties 1), 3) follow immediately from the definition of $\psi$. For 2) we use that in $[t_0, 1]$

hence

where in the last inequality we used that $W'$ is an increasing function.

□

Proof of Step 1. We use Lemma 4.2 to define

and let

Notice that $u = \Psi$ on $\partial D$, hence the minimality of $J$ implies

We decompose $D$ as the disjoint union

and notice that

where we have used that

and that $W$ is convex on its positivity set.

Combining (4.3) and (4.4) we find

In $D_1$ we use the Cauchy-Schwartz inequality and the coarea formula to obtain

On the other hand $|\nabla \Psi| = \sqrt{W(\Psi)}$ in $D_1$ by construction (see 1) in Lemma 4.2 and (2.5) and the inequality above becomes an equality for $\Psi$:

Next we use that

and the isoperimetric inequality implies

hence

We combine this with (4.5), (4.7) and use that

and obtain

The inequality holds also when we replace $\Psi$ by $\Psi_t$ defined as

Notice that $\Psi_0 = \Psi$ and $\{\Psi_t = s \}$ is the sphere at distance $t$ from the sphere $\{\Psi = s\}$. Thus, if we write the inequality above for $t \in [0, t_0]$ and average it over this interval we obtain

Let $s_1 \in [0, s_0]$ and denote by

hence if $s \le s_1$ then

Notice that by the choice of $c_\gamma$ we have

Since $W(u) \ge W(s_1)$ in the set $\{0 < u\le s_1\}$, we can bound below the left hand side in (4.11) by

while the right hand side in (4.11) is bounded above by

with $C_2$, $c_1$ universal constants.

We choose $s_1$ such that

Using that $c_\gamma \sim (2-\gamma)^2$ we find that the coefficient

which appears in (4.12) remains bounded below as $\gamma \to 2^-$. This means that if $a(1-2t_0) \le a(1) \le c_0$ small, universal, then the expression in (4.12) is decreasing in the variable $b \in [0, a(1-2t_0)]$ and is bounded below by $C_3 \, a(1-2t_0)$. In conclusion

or equivalently,

which, using that $C_3 \gg C_2$ and $t_0$ is sufficiently small, gives (4.2):

and Step 1 is proved. □

As we iterate Step 1 we find that the densities of the positivity set in $B_r$, $a(r) r^{-n}$, tend to $0$ as $r = r_0^m \to 0$. After rescaling, it remains to show that if $a(1)$ is sufficiently small, depending on $\gamma$, then $a(1/2) = 0$.

Step 2: If $a(1) \le c(\gamma)$ small then for all $r \in [1/2, 1]$,

with $\delta$, $\mu$ universal constants.

Proof of Step 2. Assume for simplicity that $r = 1$. Notice that by Theorem 2.1 it follows that

We argue as in Step 1 and improve the last part of the argument. Take

with $t_1 \in (0, t_0]$ such that

This means that $\{u < \Psi \}$ on $\partial B_1$ and now we may take $D = \{u > \Psi \} \cap B_1$. We obtain as above the corresponding inequality (4.10) with $t_0$ replaced by $t_1$. After averaging over the family of translates $\Psi_t$ with $t \in [0, t_1]$ we establish the inequality (4.11) with $t_0$ replaced by $t_1$. We bound the left hand side as before by taking

and obtain

Using that

the coefficient of $b$ in the left hand side is bounded below by a negative power of $a(1-2t_1)$ (provided that $a(1)$ is sufficiently small, depending on $\gamma$). Then, by arguing that

we obtain that the left hand side is bounded below by

for some $\delta$ universal. After relabeling $\delta$ if necessary we reach the desired discrete differential inequality claimed in Step 2.

□

End of the proof: Now it is straightforward to check that a nondecreasing function $a(r)$ that satisfies (4.13) must vanish when $r = 1/2$ if $a(1)$ is sufficiently small. In the continuous setting we obtain $a' \ge a^{1-\delta}$ which implies

for some large $M$, provided that the inequality is satisfied at $r = 1$. In the discrete setting it follows by induction that the inequality above holds for $r = r_k$ where $r_k$ is the sequence

□

Remark 4.3 From (4.5) and (4.7) it follows that

with $C$ universal.

Next we prove the other side of the density bound using a similar analysis.

Lemma 4.4. Let $u$ be a minimizer of $J_\gamma$ in $B_1$ and assume $0\in F(u).$ Then

Proof. Let $s_0$, $s_1$, and $t_1$ be defined as

with $M$ a large universal constant to be made precise later. Let

Step 1: We prove that if $a(1) \le c_0$ universal, $M \ge C_0$ and $\gamma$ sufficiently close to $2$ (depending on $M$) then

We first construct a 1D profile.

Lemma 4.5. There exists a nondecreasing Lipschitz function $\psi:[0, 1] \to \mathbb R$, with $\psi(0) = 0$, which is $C^1$ in the intervals $\{\psi < s_1\}$, $\{\psi > s_1\}$ such that

1) $\psi = \varphi$ in $[0, t_1] = \{ \psi \le s_1\}$,

2)

and $\psi$ is constant in $[1/4, 1]$,

3)

Here $t_0$ is defined such that

Proof of Step 1. Define in $\overline B_1$ the function

and denote by

Notice that $\Psi$ vanishes on $\partial B_1$ and coincides with $\varphi(1-|x|)$ near $\partial B_1$, hence

Also by 2)

and 3) implies

and

Denote by

Then $J(u, D) \le J(\Psi, D)$ implies

In

we write

and notice that $w$ vanishes on $\partial F_1$ hence

where in the last inequality we used the convexity of $W$ in $D_1$ and the fact that $W'(\Psi) < 0$ in $A_{1-t_1}$ thus

Since $\Psi-w = u$ in $D_1$ we find

hence

By Cauchy-Schwartz and co-area formula we obtain

while, by (4.15),

Hence

and we also write

By (4.17) the second term is bounded by $|A_1|$, while by (4.16) and the co-area formula as above, the first integral is bounded by

Using that

we find by the isoperimetric inequality that

Notice that as $\gamma \to 2$ (and fixed $M$), the integral converges to

Also

In conclusion

Since $|E| \le a(1)\le c_0$ is sufficiently small, and $M \ge C_0$, the left hand side is bounded below by

We average the right hand side by taking as test functions

and obtain

which, as in the proof of Lemma 4.2, implies the desired conclusion (4.14) with $r_0 = 1-2t_0$,

provided $C_0$ is chosen sufficiently large.

□

Next we prove the lemma when $\gamma$ is close to 2.

Step 2: If $\gamma$ is sufficiently close to 2 then $|\{u = 0\} \cap B_1| \ge c_0/2$.

Proof of Step 2. If the conclusion does not hold then

Indeed, otherwise

and we can apply inequality (4.19) (with $A_{1-t_1}$, $A_1$ replaced by $A_1$, respectively $A_{1+t_1}$) and obtain

We get a contradiction by choosing $M$ universal, sufficiently large, and (4.20) is proved. Now we may apply Step 1 and obtain

with $\alpha$ as in (2.4), which can be rescaled and iterated indefinitely. Thus, after a rescaling of $u$ of factor $r_0^m$ with $m$ large we find that $a(1)$ can be made arbitrarily small.

We reached a contradiction to $0 \in F(u)$ since, by Theorem 2.1,

□

Finally, we prove the conclusion also when $\gamma$ stays away from 2.

Step 3: If $\gamma \le 2-\delta$ then $|\{u = 0\} \cap B_1| \ge c(\delta)$.

Proof of Step 3. This follows easily by compactness. However, here we sketch a direct proof that follows from an argument in Step 1.

First we claim that

for some $c(\delta) > 0$ small. Otherwise, the energy of $u$ in $B_{1/2}$ is sufficiently small, which implies that $\{u > 0\}$ has small measure in $B_{1/2}$ and contradicts Lemma 4.1.

Next, let $v$ be the solution to the Euler-Lagrange equation $2 \triangle \Psi = W'(\Psi)$ in $B_1$, $v = u$ on $\partial B_1$. Since $v$ is superharmonic, $v(0) > c(\delta)$. Moreover, $W(v)$ is bounded by an integrable function in $B_1$. As in Step 1, the inequality

implies (see (4.18) with $s_1 = 0$, $D_1 = F_1 = B_1$),

The left hand side is bounded below by a $c_1(\delta)$ which follows from Theorem 2.1 and $(v-u)(0) = v(0) \ge c(\delta)$. This shows that $\{u = 0\}$ cannot have arbitrarily small measure. □

It remains to prove the existence of the 1D profile of Lemma 4.5.

Proof of Lemma 4.5. We construct $\psi$ by defining its corresponding function $g$ as in $(3.2)$, $(3.3)$. Let $g$ be the perturbation of $W$

with $C_n = 8n$. Let $s_2$ be defined as

hence $s_2 = 4^{1/\gamma} s_0 \sim s_0$, and notice that $s_2 \to 0$ as $\gamma \to 2$. Moreover

which implies that $g' \le -C$ in the same interval. Furthermorer, for $\gamma$ sufficiently close to $2$ (depending on $M$), then $s_1^{1-\gamma/2}$ is close to $1$ hence the error $g(s)-W(s)$ is uniformly close to the constant $-1/2$ in the interval $[s_1, 1]$.

These facts imply that $g \le W$, and $g$ crosses $0$ at some point $\sigma \in [s_0, s_2]$, and

which gives property 3). Property 1) follows directly from the definition. Finally, property 2) holds since in $(s_1, 1] \cap \{g > 0\}$

Moreover,

which shows that $\psi$ is constant outside an interval of length $1/4$. □

We conclude this section with a proof of Corollary 2.3.

Proof of Corollary 2.3. Assume that $u$ is a minimizer of $J$ in $B_2$ and $0 \in F(u)$. First we prove that

with $c$, $C$ universal constants.

The upper bound follows from Remark 4.3. For the lower bound, we use that

Let $s_0$ be defined as in the proof of Lemma 4.1, see (4.1). If

then

In this last set $W(u) \ge W(s_0) = 1$, and the lower bound is obtained from the potential term.

On the other hand, if the opposite inequality in (4.22) holds, then for all $s \in (0, s_0)$ the density of $\{u > s\}$ in $B_1$ is bounded both above and below by universal constants. Now the lower bound follows from (4.6) and the Poincaré inequality for $\chi_{\{u > s\}}$ in $B_1$.

The existence of a full ball of radius $c'$ included in $\{u > 0\} \cap B_1$ (or $\{u = 0\}\cap B_1$) follows by a standard covering argument. We sketch it below.

We take a collection of $m$ disjoint balls $B_\rho(x_i)$, $x_i \in \{u > 0\} \cap B_1$ such that $\cup B_{5\rho}(x_i)$ covers $\{u > 0\} \cap B_1$. It follows that $m \sim \rho^{-n}$. If we assume that each $B_{\rho/2}(x_i)$ intersects the free boundary then, by the rescaled version of (4.21),

with $\alpha$ as in (2.4). We obtain

and we contradict the upper bound if $\rho$ is chosen small, universal. □

5.   The Gamma convergence

In this section we prove our main result Theorem 2.5. We start by constructing an interpolation between two functions which are close to each other in a ring.

Proposition 5.1. Let $u_k, v_k$ be sequences in $H^1(B_1)$ and $\gamma_k \to 2^-$. Assume that for some $\rho \in (\frac 12, 1)$ and $\delta > 0$ small,

are uniformly bounded, and

Then, there exists $w_k \in H^1(B_1)$ with

such that

with $o(1) \to 0$ as $k \to \infty$.

Proof. Fix $\epsilon > 0$ small. We prove the conclusion with $o(1)$ replaced by $C \epsilon$ for some $C$ universal. Since the energies of $u_k$ and $v_k$ are uniformly bounded, we can decompose the annulus $B_{\rho+ \delta}\setminus B_\rho$ into a disjoint union of $\sim \epsilon^{-1}$ annuli, and after relabeling $\rho$ and $\delta$ we may assume that

For simplicity of notation we drop the subindex $k$.

First we prove the result under the additional assumption

Denote by

and let

Notice that

Let

then, by the property (2.5) of the one-dimensional solution $\varphi$, we find

Notice that

Thus, we average (5.2) for $r \in [\rho, \rho+\delta/4],$ and obtain

We use (2.3) and the change of coordinates

The right hand side in (5.3) equals

Thus, for all $k$ sufficiently large, we can find an $r = r_k \in [\rho, \rho+\delta/4],$ such that

Since in the annulus $B_{\rho+\delta}\setminus B_{\rho}$ the function $\Psi_r$ coincides with $u$ or $v$ outside $D_r$ we find

Finally we define

with $\eta \in C_0^\infty(B_{\rho + \delta})$ a cutoff function with $\eta = 1$ in $B_{\rho+\delta/2}$. Clearly, $w = u$ outside $B_{\rho+\delta}$ and $w = \Psi_r$ in $B_{\rho+\delta /2}$, hence $w = v$ in $B_\rho$. Moreover,

Since

we find

Using that,

we obtain

We find

for all large $k$, which gives the desired conclusion under the assumption (5.1).

The general case follows easily from the interpolation procedure between the two ordered functions described above. We apply it two times, first in the annulus $B_{\rho + \delta} \setminus B_{\rho + \delta/2}$ where we interpolate between $u$ and $\min\{u, v\}$ and then in the annulus $B_{\rho + \delta/2} \setminus B_{\rho}$ where we interpolate between $\min\{u, v\}$ and $v$. □

We recall now the functional $\mathcal F$ introduced in Section 2, which is defined on the space of pairs $(u, E) \in \mathcal A(\Omega)$

given by the Dirichlet - perimeter energy

Here $P_{\Omega}(E)$ represents the perimeter of $E$ in $\Omega$

In the next two lemmas we establish the $\Gamma$-convergence of the $J_\gamma$ to $\mathcal F$.

Lemma 5.2 (Lower semicontinuity). Let $\gamma_k \to 2^-$ and $u_k$ satisfy

Then

Proof. After passing to a subsequence we may assume that the two convergences above hold pointwise a.e. in $\Omega$. This implies that $\{u > 0\} \setminus E^c$ is a set of measure zero, hence $u = 0$ a.e. on $E$, and $(u, E)$ is an admissible pair.

We write

By the coarea formula and the definition of $W$ (see (2.1))

Moreover, $\overline u_k^{1-\gamma_k/2}$ converges in $L^1$ to $\chi_E$, hence

by the lower semicontinuity of the BV norm. On the other hand

and since $(u_k - \epsilon)^+ \to (u- \epsilon)^+$ in $L^2$, we obtain

By adding the inequalities we find

and the conclusion is proved by letting $\epsilon \to 0$. □

Lemma 5.3. Let $(u, E) \in \mathcal A(\Omega)$ with $u$ a continuous function in a Lipschitz domain $\overline \Omega$. Then, given a sequence $\gamma_k \to 2^-$ we can construct a sequence $u_k$ such that

In view of the lower semicontinuity property in $\Omega \setminus \overline D$, where $D \subset \Omega$ is a subdomain, we obtain that

Proof. For the convergence of the energies it suffices to show that

Fix $\epsilon > 0$ small. First we approximate $E$ in $\Omega$ by a smooth set $F \subset \mathbb R^n$ which is included in the open set $\{u < \epsilon\}$ in $\Omega$ (which contains a neighborhood of $E$). Precisely, (see Lemma 1 of Modica ^[12]), there exists a smooth set $F \subset \mathbb R^n$ which approximates $E$ in $\Omega$ in the sense that

In view of this, it suffices to prove the lemma with $E$ replaced by $\tilde E: = F$ and $u$ replaced by $\tilde u: = (u-2 \epsilon)^+$ which is an approximation of $u$ in $H^1(\Omega)$. Notice that by construction $\tilde u$ vanishes in a $\delta$-neighborhood of $\tilde E$ for some small $\delta$. We define $u_k$ in $B_1$ as

where $d$ represents the distance in $\mathbb R^n$ to $\tilde E$. Next we check that $u_k$ satisfies the desired conclusions.

Clearly $u_k = 0$ on $\tilde E$, and using that

and $1-\gamma_k/2 \to 0^+$ we have

hence

Here we used that $\varphi_k(d) = c_\gamma^* d^\alpha$, with $c_\gamma^*$ defined in (2.4), and we have

Since $\varphi_k(d)$ converges uniformly to $0$ as $k \to \infty$ we also obtain

Using property (2.5) we obtain that

with

Notice that

represents the measure of the one-dimensional solution which, as $k \to \infty$, converges weakly in any bounded interval $[-a, a]$ to the Dirac delta measure at $0$. On the other hand (5.6) implies that

In conclusion, we find that as $k \to \infty$

and

Using that

and

we find

□

We are finally ready to prove our main theorem.

Proof of Theorem 2.5. The $L^2$ convergence follows from the uniform bound of the $u_k$ in $H^1(\Omega)$.

By the coarea formula (see (5.5)) we find that

and using the inequality

we find that $u_k^{1-\gamma_k/2}$ are uniformly bounded in $BV (\Omega)$. Thus, after passing to a subsequence, we have

for some non-negative $g \in BV(\Omega)$. We claim that

First we show that for all $\delta > 0$ small

Otherwise, for all large $k$, the set

has measure bounded below by a fixed positive constant. On this set

as $k \to \infty$ and we contradict the uniform upper bound for the energy of $u_k$ in $\Omega$.

Similarly we find that the set

Indeed, otherwise

has measure bounded below by a fixed positive constant. Then we contradict the uniform upper bound for the $L^2$ norm of $u_k$ since on the set above

as $k \to \infty$, and the claim (5.8) is proved.

The argument above implies also that

For example if

for some positive constant $\mu$ independent of $k$, then (5.7) and (5.8) imply

and we get a contradiction as in (5.9). Also

as $k \to \infty$, follows from the convergence (5.7) and (5.8).

Next we assume that $u_k$ are minimizers for $J_{\gamma_k}$ and prove the minimality of $(u, E)$ for $\mathcal F$. The argument is standard and follows from Proposition 5.1. We sketch it for completeness.

For simplicity let $\Omega = B_1$. Since the functions $u_k$ are uniformly Hölder continuous on compact sets of $B_1$ we find that the limiting function $u$ is Hölder continuous in $B_1$ and the convergence $u_k \to u$ is uniform on compact subsets.

Let $(v, F)$ be an admissible pair which coincides with $(u, E)$ near $\partial B_1$ and let

be an annulus near $\partial B_1$ where the two pairs coincide.

Denote by $v_k$ be the functions constructed in Lemma 5.3 corresponding to the pair $(v, F)$ in $B_{\rho+\delta}$. Since $u_k$ and $v_k$ satisfy the hypotheses of Proposition 5.1 we can construct $w_k$ as the interpolation between $u_k$ and $v_k$. By the minimality of $u_k$ in $B_1$ and the conclusion of Proposition 5.1 we have

This gives

and by taking $k \to \infty$, we find from Lemmas 5.2 and 5.3

We let $\rho \to 1$ and obtain the desired conclusion

Finally, the uniform convergence of the free boundaries follows from the uniform density estimates and the $L^1$ convergence (5.10). □

Acknowledgments

Daniela De Silva is supported by NSF Grant DMS 1937254. Ovidiu Savin is supported by NSF Grant DMS 2055617.

Conflict of interest

The authors declare no conflict of interest.