Regularity of the Flat Free Boundaries

Abstract

This chapter is dedicated to the regularity of the flat free boundaries. In particular, we will show how the improvement of flatness (proved in previous section) implies the C ^{1, α} regularity of the free boundary (see Fig. 8.1). The results of this section are based on classical arguments and are well-known to the specialists in the field. The main result of the chapter is the following.

This chapter is dedicated to the regularity of the flat free boundaries. In particular, we will show how the improvement of flatness (proved in previous section) implies the C ^{1, α} regularity of the free boundary (see Fig. 8.1). The results of this section are based on classical arguments and are well-known to the specialists in the field. The main result of the chapter is the following.

Fig. 8.1

A flat free boundary

FormalPara Theorem 8.1 (ε-Regularity for Viscosity Solutions)

There are dimensional constants ε > 0 and δ > 0 such that the following holds:

Suppose that $u:B_1\to \mathbb {R}$ satisfies the following conditions:

• (a) u is a non-negative continuous function and a viscosity solution of (7.1) in B ₁;

• (b) u is ε-flat in B ₁ , that is,

  $$\displaystyle \begin{aligned} (x_d-{\varepsilon})_+\le u(x)\le (x_d+{\varepsilon})_+\qquad \mathit{\text{for every}}\qquad x\in B_1. \end{aligned}$$

Then, there is α > 0 such that the free boundary ∂ Ω _u is C ^{1, α} regular in the cylinder $B_{\delta }^{\prime }\times (-\delta ,\delta )$ . Precisely, there is a function $g:B_\delta ^{\prime }\to (-\delta ,\delta )$ such that:

• (i) g is C ^{1, α} regular in the (d − 1)-dimensional ball $B_\delta ^{\prime }\subset \mathbb {R}^{d-1}$;

• (ii) the set $\Omega _u\cap \big (B_\delta ^{\prime }\times (-\delta ,\delta )\big )$ is the supergraph of g, that is,

  $$\displaystyle \begin{aligned} \displaystyle\Omega_u\cap \big(B_\delta^{\prime}\times (-\delta,\delta)\big)=\big\{x=(x',x_d)\in B_\delta^{\prime}\times (-\delta,\delta)\ :\ x_d>g(x')\big\}. \end{aligned}$$

Moreover, g (and so, ∂ Ω _u ) is C ^{1, α} regular, for any α ∈ (0, 1∕2).

FormalPara Proof

The existence of a function $g: B_\delta ^{\prime }\subset \mathbb {R}^{d-1}$, which is C ^{1, α} regular, for some α > 0, for which (ii) holds, is a consequence of:

• Theorem 7.4, in which we show that the improvement of flatness (Condition 8.3) holds for viscosity solutions (with constants σ = C _d κ);

• Lemma 8.4, in which we show that the improvement of flatness implies the uniqueness of the blow-up limit and the decay of the blow-up sequence:

  $$\displaystyle \begin{aligned} \|u_{r,x_0}-u_{x_0}\|{}_{L^\infty(B_1)}\le C_dr^\gamma\quad \text{for every}\quad r<{1}/{2}\quad \text{and every}\quad x_0\in B_{{1}/{2}}, \end{aligned} $$

  (8.1)

  where γ is such that κ ^γ = σ;

• Proposition 8.6, in which we show that if (8.1) holds, then ∂ Ω_u is C ^{1, α} regular in B _1∕2, where $\alpha =\frac {\gamma }{1+\gamma }$.

In particular, we notice that by choosing κ small enough, we can take γ as close to 1 (and so, α as close to 1∕2) as we want. □

As a consequence, we obtain the regularity of the free boundary for minimizers of $\mathcal F_\Lambda$.

FormalPara Corollary 8.2 (Regularity of Reg(∂ Ω_u))

Let D be a bounded open set in $\mathbb {R}^d$ and let $u:D\to \mathbb {R}$ be a (non-negative) minimizer of $\mathcal F_\Lambda$ in D. Then, every regular point x ₀ ∈ Reg(∂ Ω _u) ⊂ D has a neighborhood $\mathcal U$ such that $\partial \Omega _u\cap \mathcal U$ is a C ^{1, α} regular manifold, for every α ∈ (0, 1∕2).

FormalPara Proof

Notice that, up to replacing u(x) by v(x) = Λ^−1∕2 u(x), we may assume that Λ = 1. By the definition of Reg(∂ Ω_u) (see Sect. 6.4), there is a sequence r _n → 0 such that the blow-up sequence $u_{r_n,x_0}$ converges uniformly (in B ₁) to a function $u_0:\mathbb {R}^d\to \mathbb {R}$ of the form

$$\displaystyle \begin{aligned} u_0(x)=\,(x\cdot\nu)_+ \end{aligned}$$

for some unit vector $\nu \in \mathbb {R}^d$. Then, by Proposition 6.2, for n large enough, we have

$$\displaystyle \begin{aligned} \|u_{r_n,x_0}-u_0\|{}_{L^\infty(B_1)}<{\varepsilon}, \end{aligned}$$

$$\displaystyle \begin{aligned} u_{r_n,x_0}>0\quad \text{in}\quad \{x\cdot\nu>{\varepsilon}\}\qquad \text{and}\qquad u_{r_n,x_0}=0\quad \text{in}\quad \{x\cdot\nu<-{\varepsilon}\}. \end{aligned}$$

This means that $u_{r_n,x_0}$ is 2ε-flat in B ₁, that is,

$$\displaystyle \begin{aligned} \big(x\cdot\nu-2{\varepsilon}\big)_+\le u_{r_n,x_0}(x)\le \big(x\cdot\nu+2{\varepsilon}\big)_+\qquad \text{for every}\qquad x\in B_1. \end{aligned}$$

Now, taking ε small enough and applying Theorem 7.4, Proposition 7.1 and Theorem 8.1, we get the claim. □

This chapter is organized as follows.

In Sect. 8.1, we prove that the improvement of flatness (Condition 8.3) implies the uniqueness of the blow-up limit and gives a (polynomial) rate of convergence of the blow-ups in L ^∞(B ₁).

In Sect. 8.2, we prove that the uniqueness of the blow-up limit and the polynomial rate of convergence of the blow-up sequence imply the regularity of the free boundary. We notice that the uniqueness of the blow-up limit and the rate of convergence of the blow-up sequence can be obtained also by different arguments, for instance, via an epiperimetric inequality. In fact, the result of this section can be used also in combination with Theorem 12.1, which is an alternative way to the regularity of the free boundary.

8.1 Improvement of Flatness, Uniqueness of the Blow-Up Limit and Rate of Convergence of the Blow-Up Sequence

Condition 8.3 (Improvement of Flatness)

Let $u:B_1\to \mathbb {R}$ be a non-negative function. There are constants κ ∈ (0, 1), σ ∈ (0, 1), C ₀ > 0 and ε ₀ > 0 such that:

$$\displaystyle \begin{gathered} \mathit{For\ every}\ x_0\in \partial\Omega_u\cap B_1, r\le \mathit{\text{dist}}(x_0,\partial B_1)\ \mathit{and}\ {\varepsilon}\in(0,{\varepsilon}_0]\ \mathit{satisfying}\\ (x\cdot\nu-{\varepsilon})_+\le u_{r,x_0}\le (x\cdot\nu+{\varepsilon})_+\quad \mathit{\text{in}}\quad B_1,\\ \mathit{there\ is}\ \tilde \nu\in\partial B_1\ \mathit{such\ that}\\ |\tilde\nu-\nu|\le C_0 {\varepsilon}\qquad \mathit{\text{and}}\qquad \big(x\cdot\tilde \nu-\sigma{{\varepsilon}}\big)_+\le u_{\kappa r,x_0}\le \big(x\cdot\tilde \nu+\sigma{{\varepsilon}}\big)_+\quad \mathit{\text{in}}\quad B_1. \end{gathered} $$

Lemma 8.4 (Uniqueness of the Blow-Up Limit)

Suppose that $u:B_1\to \mathbb {R}$ is a continuous non-negative function satisfying Condition 8.3. Then, there are constant ε ₁ > 0, γ > 0 and C ₁ > 0 (depending on ε ₀ , κ, σ and C ₀ ) such that if

$$\displaystyle \begin{aligned} (x\cdot\nu-{\varepsilon}_1)_+\le u\le (x\cdot\nu+{\varepsilon}_1)_+\quad \mathit{\text{in}}\quad B_1, \end{aligned}$$

for some ν ∈ ∂B ₁ , then for every x ₀ ∈ ∂ Ω _u ∩ B _1∕2 there is a unique unit vector

$$\displaystyle \begin{aligned} \nu_{x_0}\in\partial B_1\subset\mathbb{R}^d \end{aligned}$$

such that

$$\displaystyle \begin{aligned} \|u_{r,x_0}-u_{x_0}\|{}_{L^\infty(B_1)}\le C_1r^\gamma\qquad \mathit{\text{for every}}\qquad r\le{1}/{2}\,, \end{aligned}$$

where the function $u_{x_0}$ is defined as

$$\displaystyle \begin{aligned} u_{x_0}(x)=(\nu_{x_0}\cdot x)_+\qquad \mathit{\text{for every}}\qquad x\in \mathbb{R}^d. \end{aligned}$$

Precisely, we can take γ, ε ₁ and C ₁ as follows:

$$\displaystyle \begin{aligned} {\varepsilon}_1=\frac{{\varepsilon}_0}4\ ,\qquad \kappa^\gamma=\sigma\qquad \mathit{\text{and}}\qquad C_1=\big(\frac2\kappa\big)^\gamma \left(1+\frac{C_0}{1-\sigma}+\frac{1}\kappa\right){\varepsilon}_0. \end{aligned}$$

Proof

Let ${\varepsilon }_1=\frac {{\varepsilon }_0}4.$ Notice that if u is ε ₁-flat in B ₁, then

$$\displaystyle \begin{gathered} u_{{1}/{2},x_0}\ \text{is}\ {\varepsilon}_0\text{-flat in}\ B_1, \end{gathered} $$

for every x ₀ ∈ ∂ Ω_u ∩ B _1∕2.

Let x ₀ ∈ ∂ Ω_u ∩ B _1∕2 be fixed,

$$\displaystyle \begin{aligned} \displaystyle r_n=\frac{\kappa^n}{2}\qquad \text{and}\qquad u_n:=u_{r_n,x_0}. \end{aligned}$$

By the improvement of flatness, there is a sequence of unit vectors ν _n ∈ ∂B ₁ such that

$$\displaystyle \begin{aligned} \big(x\cdot \nu_n-{{\varepsilon}_0}{\sigma^n}\big)_+\le u_{n}\le \big(x\cdot \nu_n+{{\varepsilon}_0}{\sigma^n}\big)_+\quad \text{in}\quad B_1\, , \end{aligned}$$

and

$$\displaystyle \begin{aligned} |\nu_n-\nu_{n+1}|\le C_0{{\varepsilon}_0}{\sigma^n}\qquad \text{for every}\qquad n\in\mathbb{N}. \end{aligned}$$

In particular, for every 1 ≤ n < m, we have

$$\displaystyle \begin{aligned} |\nu_n-\nu_m|\le \sum_{k=n}^{m-1}|\nu_k-\nu_{k+1}|\le \sum_{k=n}^{m-1}C_0{{\varepsilon}}{\sigma^k}\le {\varepsilon} C_0\sum_{k=n}^\infty\sigma^k= \frac{C_0{\varepsilon}}{1-\sigma}\sigma^n. \end{aligned}$$

This implies that there is a vector ν _∞∈ ∂B ₁ such that

$$\displaystyle \begin{aligned} \nu_\infty=\lim_{n\to\infty}\nu_n\qquad \text{and}\qquad |\nu_n-\nu_\infty|\le \sum_{k=n}^\infty|\nu_k-\nu_{k+1}|\le \frac{C_0{\varepsilon}_0}{1-\sigma}\sigma^n. \end{aligned}$$

Thus,

$$\displaystyle \begin{aligned} \big|x\cdot\nu_\infty-\big(x\cdot\nu_n\pm{{\varepsilon}_0}{\sigma^n}\big)\big|\le \left(1+\frac{C_0}{1-\sigma}\right){\varepsilon}_0\sigma^n\qquad \text{for every}\qquad x\in B_1, \end{aligned}$$

which implies that

$$\displaystyle \begin{aligned} |(x\cdot\nu_\infty)_+-u_n(x)|\le\left(1+\frac{C_0}{1-\sigma}\right){\varepsilon}_0\sigma^n\qquad \text{for every}\qquad x\in B_1. \end{aligned}$$

Now, we set

$$\displaystyle \begin{aligned} u_{0}(x)=(x\cdot\nu_\infty)_+. \end{aligned}$$

Let r ≤ 1∕2 be arbitrary and let $n\in \mathbb {N}$ be such that

$$\displaystyle \begin{aligned} r_{n+1}=\frac 12\kappa^{n+1}<r\le\frac 12\kappa^n=r_n. \end{aligned}$$

Then, there is ρ ∈ (κ, 1] such that r = ρr _n. Since $u_{r_n,x_0}$ satisfies

$$\displaystyle \begin{aligned} \big(x\cdot \nu_n-{{\varepsilon}_0}{\sigma^n}\big)_+\le u_{r_n,x_0}(x)\le \big(x\cdot \nu_n+{{\varepsilon}_0}{\sigma^n}\big)_+\qquad \text{in}\qquad B_1\, , \end{aligned}$$

we get that $u_{r,x_0}=u_{\rho r_n,x_0}$ satisfies

$$\displaystyle \begin{aligned} \big(x\cdot \nu_n-\frac{{\varepsilon}_0}{\rho}\sigma^n\big)_+\le u_{r,x_0}\le \big(x\cdot \nu_n+\frac{{\varepsilon}_0}{\rho}\sigma^n\big)_+\qquad \text{in}\qquad B_1\, , \end{aligned}$$

which implies that

$$\displaystyle \begin{aligned} \|u_{r_n,x_0}-u_{r,x_0}\|{}_{L^\infty(B_1)}& \le \frac{{\varepsilon}_0}{\rho}\sigma^n\le \frac{{\varepsilon}_0}{\kappa}\sigma^n, \end{aligned} $$

and finally gives that

$$\displaystyle \begin{aligned} \|u_{r,x_0}-u_0\|{}_{L^\infty(B_1)}\le \left(1+\frac{C_0}{1-\sigma}+\frac{1}\kappa\right){\varepsilon}_0\sigma^{n}. \end{aligned}$$

Since κ ^γ = σ, we get that

$$\displaystyle \begin{aligned} \sigma^n\le \kappa^{n\gamma}\le \frac 1{\kappa^\gamma}\big(\kappa^{n+1}\big)^\gamma\le \frac 1{\kappa^\gamma}\big(2r\big)^\gamma=\big(\frac2\kappa\big)^\gamma r^\gamma, \end{aligned}$$

from which, we deduce

$$\displaystyle \begin{aligned} \|u_{r,x_0}-u_0\|{}_{L^\infty(B_1)}\le \big(\frac2\kappa\big)^\gamma \left(1+\frac{C_0}{1-\sigma}+\frac{1}\kappa\right){\varepsilon}_0r^\gamma, \end{aligned}$$

which concludes the proof. □

8.2 Regularity of the One-Phase Free Boundaries

Condition 8.5

(Uniqueness of the Blow-Up Limit and Rate of Convergence of the Blow-Up Sequence) The function $u:B_1\to \mathbb {R}$ satisfies this condition if it is non-negative and if there are constants C ₁ > 0 and γ > 0 such that, for every x ₀ ∈ ∂ Ω _u ∩ B _1∕2 there is a unique function $u_{x_0}:B_1\to \mathbb {R}$ such that:

• (i) there is $\nu _{x_0}\in \partial B_1$ such that $u_{x_0}(x)=(\nu _{x_0}\cdot x)_+$ for every x ∈ B ₁;

• (ii) $\|u_{r,x_0}-u_{x_0}\|{ }_{L^\infty (B_1)}\le C_1r^\gamma$ for every r ≤ 1∕2.

Proposition 8.6 (The Condition 8.5 Implies the Regularity of ∂ Ω_u)

Let $u:B_1\to \mathbb {R}$ be a non-negative function such that:

• (a) u is Lipschitz continuous on B ₁ and $L=\|\nabla u\|{ }_{L^\infty (B_1)}$;

• (b) u is non-degenerate in the sense that there is a constant η > 0 such that

  $$\displaystyle \begin{aligned} \mathit{\text{if}}\quad y_0\in\overline\Omega_u\cap\partial B_{{1}/{2}}\,,\quad \mathit{\text{then}}\quad \|u\|{}_{L^\infty(B_r(y_0))}\ge \eta r\,,\quad \mathit{\text{for every}}\quad r\in(0,{1}/{2}). \end{aligned}$$

• (c) u satisfies Condition 8.5 for some γ > 0 and C ₁ > 0.

Then, there is ρ > 0 such that ∂ Ω _u is a C ^{1, α} manifold in B _ρ , where $\alpha :=\frac {\gamma }{1+\gamma }$.

Precisely, there are ρ > 0 and a C ^{1, α} -regular function $g:B^{\prime }_\rho \to (-\rho ,\rho )$ such that, up to a rotation of the coordinate system of $\mathbb {R}^d$ , we have

Lemma 8.7 (Flatness of the Free Boundary ∂ Ω_u)

Let $u:B_1\to \mathbb {R}$ be a non-negative function such that

• (a) u satisfies the Condition 8.5 with constants C ₁ and γ.

• (b) u is non-degenerate, that is, there is a constant η > 0 such that

  $$\displaystyle \begin{aligned} \mathit{\text{if}}\quad y_0\in\overline\Omega_u\cap\partial B_{{1}/{2}}\,,\quad \mathit{\text{then}}\quad \|u\|{}_{L^\infty(B_r(y_0))}\ge \eta r\,,\quad \mathit{\text{for every}}\quad r\in(0,{1}/{2}). \end{aligned}$$

Then, there are constants C > 0 and r ₀ > 0 such that, for every x ₀ ∈ ∂ Ω _u ∩ B _1∕2 , we have

$$\displaystyle \begin{aligned} & \Omega_{{x_0,r}}\cap B_1\supset \{x\in B_1\,:\, x\cdot\nu_{x_0}>Cr^\gamma\}\\ & \quad \mathit{\text{and}}\quad \Omega_{{x_0,r}}\cap \{x\in B_1\,:\, x\cdot\nu_{x_0}<-Cr^\gamma\}=\emptyset, \end{aligned} $$

(8.2)

for every r ∈ (0, r ₀), where $\Omega _{x_0,r}:=\{u_{x_0,r}>0\}$.

Proof

In order to prove the first part of (8.2), we notice that

$$\displaystyle \begin{aligned} \|u_{r,x_0}-u_{x_0}\|{}_{L^\infty(B_1)}\le C_1r^\gamma \end{aligned}$$

implies that

$$\displaystyle \begin{aligned} u_{r,x_0}(x)\ge \big(x\cdot\nu_{x_0}-C_1r^\gamma\big)_+\qquad \text{for every}\qquad x\in B_1. \end{aligned}$$

This gives the first inclusion of (8.2) for any constant C ≥ C ₁. In order to prove the second inclusion in (8.2), we suppose that there is a point y ∈ B ₁ such that

$$\displaystyle \begin{aligned} u_{r,x_0}(y)>0\qquad \text{and}\qquad y\cdot\nu_{x_0}<-Cr^\gamma. \end{aligned}$$

This implies that $\tilde y:=\frac {y}2\in B_{{1}/{2}}$ is such that

$$\displaystyle \begin{aligned} u_{2r,x_0}(\tilde y)>0\qquad \text{and}\qquad \tilde y\cdot\nu_{x_0}<-\frac 12Cr^\gamma. \end{aligned}$$

The non-degeneracy of u now implies that

$$\displaystyle \begin{aligned} \|u_{2r,x_0}\|{}_{L^\infty(B_\rho(\tilde y))}\ge \eta\, \rho\qquad \text{where}\qquad \rho:=\frac 12Cr^\gamma. \end{aligned}$$

Notice that $u_{x_0}=0$ on $B_\rho (\tilde y)$. On the other hand, choosing r ₀ such that

$$\displaystyle \begin{aligned} Cr_0^\gamma\le 1, \end{aligned}$$

we get that ρ ≤ 1∕2 and so $B_\rho (\tilde y)\subset B_1$. Thus, we have that

$$\displaystyle \begin{aligned} \frac\eta2Cr^\gamma\le \|u_{2r,x_0}-u_{x_0}\|{}_{L^\infty(B_1)}\le C_1 (2r)^\gamma, \end{aligned}$$

which is a contradiction, if we choose

$$\displaystyle \begin{aligned} C\ge \frac{2}{\eta}C_1, \end{aligned}$$

which concludes the proof by taking

$$\displaystyle \begin{aligned} C=\big(1+\frac2\eta\big)C_1\qquad \text{and}\qquad r_0=\inf\big\{{1}/{2}\,,C^{-\gamma}\big\}. \end{aligned}$$

□

Lemma 8.8 (Oscillation of ν)

Let $u:B_1\to \mathbb {R}$ be a Lipschitz continuous function and let $L=\|\nabla u\|{ }_{L^\infty (B_1)}$ . Suppose that u satisfies the Condition 8.5 with the constants C ₁ and γ. Then, there are constants R ∈ (0, 1), α and C such that

$$\displaystyle \begin{aligned} |\nu_{x_0}-\nu_{y_0}|\le C|x_0-y_0|{}^\alpha\qquad \mathit{\text{for every}}\qquad x_0,y_0\in\partial \Omega_u\cap B_{R}\,. \end{aligned} $$

(8.3)

Precisely, one can take

$$\displaystyle \begin{aligned} C=2\sqrt{d+2}\,\big(L+2C_1\big)\ ,\qquad \alpha=\frac{\gamma}{1+\gamma}\qquad \mathit{\text{and}}\qquad R=2^{-(2+\gamma)}\,. \end{aligned}$$

Proof

Let $\alpha :=\frac {\gamma }{1+\gamma }$. Let x ₀, y ₀ ∈ B _R ∩ ∂ Ω_u and r := |x ₀ − y ₀|^1−α. Then, for every x ∈ B ₁, we have

$$\displaystyle \begin{aligned} \big|u_{x_0,r}(x) - u_{y_0,r}(x)\big| = \frac 1r\big|u(x_0+rx)-u(y_0+rx)\big|\le L\frac{|x_0-y_0|}r=L |x_0-y_0|{}^{\alpha}, \end{aligned}$$

which gives that

$$\displaystyle \begin{aligned} \|u_{x_0,r} - u_{y_0,r}\|{}_{L^\infty(B_1)} \le L |x_0-y_0|{}^{\alpha}. \end{aligned}$$

On the other hand, Condition 8.5 gives that

$$\displaystyle \begin{aligned} \|u_{x_0,r} - u_{x_0}\|{}_{L^\infty(B_1)}\le C_1r^\gamma\qquad \text{and}\qquad \|u_{y_0,r} - u_{y_0}\|{}_{L^\infty(B_1)} \le C_1r^\gamma. \end{aligned}$$

We notice that in order to apply Condition 8.5 we need that r ≤ 1∕2 and R ≤ 1∕2. We choose R such that (2R)^1−α ≤ 1∕2. Thus, by the triangular inequality and the fact that r ^γ = |x ₀ − y ₀|^α, we obtain

$$\displaystyle \begin{aligned} {} \|u_{x_0} - u_{y_0}\|{}_{L^\infty(B_1)} \leq \big(L +2C_1\big)|x_0-y_0|{}^{\alpha}. \end{aligned}$$

The conclusion now follows by a general argument. Indeed, for any $v_1,v_2 \in \mathbb {R}^d$, we have

$$\displaystyle \begin{aligned} \Big( \frac{\omega_d}{d+2} \Big)^{{1}/{2}} |v_1 & - v_2| = \left(\int_{B_1} |v_1 \cdot x - v_2 \cdot x|{}^2 \,dx \right)^{{1}/{2}} \\ & \le \left( \int_{B_1} |(v_1 \cdot x)_+ - (v_2 \cdot x)_+|{}^2 \,dx \right)^{{1}/{2}} + \left(\int_{B_1} |(v_1 \cdot x)_- - (v_2 \cdot x)_-|{}^2 \,dx \right)^{{1}/{2}}\\ & = 2 \left(\int_{B_1} |(v_1 \cdot x)_+ - (v_2 \cdot x)_+|{}^2 \,dx \right)^{{1}/{2}}\le 2\omega_d^{{1}/{2}}\|(v_1 \cdot x)_+ - (v_2 \cdot x)_+\|{}_{L_x^\infty(B_1)}, \end{aligned} $$

which implies that

$$\displaystyle \begin{aligned} |v_1 - v_2| \le 2\sqrt{d+2}\, \|(v_1 \cdot x)_+ - (v_2 \cdot x)_+\|{}_{L_x^\infty(B_1)}. \end{aligned}$$

Applying the above estimate to $v_1=\nu _{x_0}$ and $v_2=\nu _{y_0}$, we get (8.3). □

Proof of Proposition 8.6

We first notice that, for every ε > 0, there exists R > 0 such that, for x ₀ ∈ ∂ Ω_u ∩ B _R we have

$$\displaystyle \begin{aligned} \begin{cases} u>0\,\text{ on }\, \mathcal C_\varepsilon^+(x_0,\nu_{x_0}) \cap B_R(x_0),\\ u=0 \,\text{ on }\, \mathcal C_\varepsilon^-(x_0,\nu_{x_0}) \cap B_R(x_0), \end{cases} \end{aligned} $$

(8.4)

where for a vector ν ∈ ∂B ₁, we denote by $\mathcal C_\varepsilon ^+(x_0,\nu )$ and $\mathcal C_\varepsilon ^-(x_0,\nu )$ the cones

$$\displaystyle \begin{aligned} \mathcal C^\pm_\varepsilon(x_0,\nu) := \left\{ x \in \mathbb{R}^d\ :\ \pm\, \nu \cdot (x-x_0) > \varepsilon |x-x_0| \right\} \end{aligned}$$

(see Fig. 8.2).

Fig. 8.2

The sets $\mathcal C^\pm _\varepsilon (x_0,\nu )$

Indeed, the flatness estimate (8.2) implies (8.4) by taking R such that CR ^γ ≤ ε, where C and γ are the constants from Lemma 8.7.

Let ν ₀ be the normal vector at the origin 0 ∈ ∂ Ω_u. Without loss of generality we can suppose that ν ₀ = e _d. In particular, if u ₀(x) = (x ⋅ ν ₀)₊ is the blow-up limit in zero, then

$$\displaystyle \begin{aligned} \Omega_{u_{0}}=\{u_{0}>0\}=\big\{(x',x_d)\in\mathbb{R}^{d-1}\times\mathbb{R}\,:\,x_d>0\big\}. \end{aligned}$$

Let ε ∈ (0, 1) and R > 0 be as in (8.4) and set

$$\displaystyle \begin{aligned} \rho = R\sqrt{1-{\varepsilon}^2}\qquad \text{and}\qquad \ell={\varepsilon} R. \end{aligned}$$

Let $x'\in B_\rho ^{\prime }$. Then, by (8.4), we have:

• the vertical section

  $$\displaystyle \begin{aligned} \mathcal S_+^{x'}:=\{(x',t)\in B_R \ :\ u(x',t)>0\} \end{aligned}$$

  contains the segment

  $$\displaystyle \begin{aligned} \{(x',t)\in B_R \ :\ t>{\varepsilon} R\}; \end{aligned}$$

• the closed set

  $$\displaystyle \begin{aligned} \mathcal S_0^{x'}:=\{(x',t)\in B_R \ :\ u(x',t)=0\} \end{aligned}$$

  contains the segment

  $$\displaystyle \begin{aligned} \{(x',t)\in B_R \ :\ t<-{\varepsilon} R\}. \end{aligned}$$

This implies that the function

$$\displaystyle \begin{aligned} g(x') := \inf \big\{ t \in \mathbb{R} : u(x',T)>0\ \text{for every}\ T\in(t,\rho)\big\}, \end{aligned}$$

is well defined for $x'\in B_\rho ^{\prime }$ (see Fig. 8.3).

Fig. 8.3

Graphicality of the free boundary

Let δ ≤ ρ. Let $x_0^{\prime }\in B_\delta ^{\prime }$ and let $t_0:=g(x_0^{\prime })$. By definition, we have

$$\displaystyle \begin{aligned} x_0:=(x_0^{\prime},t_0)\in\partial\Omega_u\cap B_{R}. \end{aligned}$$

Moreover, by construction, we have

$$\displaystyle \begin{aligned} -{\varepsilon} |x_0^{\prime}|\le g(x_0^{\prime})\le {\varepsilon} |x_0^{\prime}|. \end{aligned}$$

Thus,

$$\displaystyle \begin{aligned} |x_0|\le \delta\sqrt{1+{\varepsilon}^2}\le \sqrt{ 2}\,\delta. \end{aligned}$$

We next claim that, for δ small enough, we have that

$$\displaystyle \begin{aligned} u>0\,\text{ on }\, \mathcal C_{2\varepsilon}^+(x_0,e_d) \cap B_{R}(x_0)\, \qquad \text{and}\qquad u=0 \,\text{ on }\, \mathcal C_{2\varepsilon}^-(x_0,e_d) \cap B_{R}(x_0). \end{aligned} $$

(8.5)

Indeed, applying (8.4) for the point x ₀, we have

$$\displaystyle \begin{aligned} {} u>0\,\text{ on }\, \mathcal C_{\varepsilon}^+(x_0,\nu_{x_0}) \cap B_{R}(x_0)\, \qquad \text{and}\qquad u=0 \,\text{ on }\, \mathcal C_{\varepsilon}^-(x_0,\nu_{x_0}) \cap B_{R}(x_0), \end{aligned}$$

so, it is sufficient to prove that

$$\displaystyle \begin{aligned} \mathcal C_{2\varepsilon}^\pm(x_0,e_d)\subset \mathcal C_{\varepsilon}^\pm(x_0,\nu_{x_0}). \end{aligned}$$

Let $x\in \mathcal C_{2\varepsilon }^\pm (x_0,e_d)$. Then,

$$\displaystyle \begin{aligned} \nu_{x_0}\cdot (x-x_0) & = e_d\cdot (x-x_0)+(\nu_{x_0}-e_d)\cdot (x-x_0)\\ & > 2{\varepsilon}|x-x_0|-C \big(\sqrt{ 2}\,\delta\big)^\alpha |x-x_0|> \varepsilon|x-x_0|, \end{aligned} $$

where:

• for the first inequality we used the definition of $\mathcal C_{2\varepsilon }^\pm (x_0,e_d)$ and the following estimate, which is a consequence of Lemma 8.8:

  $$\displaystyle \begin{aligned} |\nu_{x_0}-e_d|\le C|x_0|{}^\alpha \le C \big(\sqrt{ 2}\,\delta\big)^\alpha\,; \end{aligned}$$

• for the second in equality, we choose δ such that $C \big (\sqrt { 2}\,\delta \big )^\alpha \le \varepsilon$.

This proves (8.5). As a consequence, we obtain that the sections $\mathcal S_+^{x'}$ and $\mathcal S_0^{x'}$ are segments:

and so, the free boundary is precisely the graph of g, that is,

$$\displaystyle \begin{aligned} \big(B^{\prime}_\delta\times(-\delta,\delta)\big)\,\cap\, \partial\Omega_u=\big\{(x',t)\in B^{\prime}_\delta\times(-\delta,\delta)\,:\, g(x')= t\big\}. \end{aligned}$$

We next prove that the function $g:B_\delta ^{\prime }\to \mathbb {R}$ is Lipschitz continuous on $B_\delta ^{\prime }$. Also this follows by the uniform cone condition (8.5). Indeed, let

$$\displaystyle \begin{aligned} x_1^{\prime},x_2^{\prime}\in B_\delta^{\prime}\ ,\quad x_{1}=(x_1^{\prime},g(x_1^{\prime}))\quad \text{and}\quad x_{2}=(x_2^{\prime},g(x_2^{\prime})). \end{aligned}$$

Since $x_1\notin \mathcal C^+_{2\varepsilon }(x_2,e_d)$, we have that

$$\displaystyle \begin{aligned} g(x_1^{\prime})- g(x_2^{\prime})=(x_1-x_2)\cdot e_d\le 2\varepsilon|x_1-x_2|\le 2\varepsilon|x_1^{\prime}-x_2^{\prime}|+2\varepsilon|g(x_1^{\prime})- g(x_2^{\prime})|. \end{aligned}$$

Analogously, $x_2\notin \mathcal C^+_{2\varepsilon }(x_1,e_d)$ implies that

$$\displaystyle \begin{aligned} g(x_1^{\prime})- g(x_2^{\prime})\le 2\varepsilon|x_1^{\prime}-x_2^{\prime}|+2\varepsilon|g(x_1^{\prime})- g(x_2^{\prime})|, \end{aligned}$$

and the two estimates give

$$\displaystyle \begin{aligned} \left(1-2\varepsilon\right)|g(x_1^{\prime})- g(x_2^{\prime})|\le 2\varepsilon|x_1^{\prime}-x_2^{\prime}|, \end{aligned}$$

and finally, choosing ε ≤ 1∕4, we get

$$\displaystyle \begin{aligned} |g(x_1^{\prime})- g(x_2^{\prime})|\le 4\varepsilon|x_1^{\prime}-x_2^{\prime}|, \end{aligned}$$

which concludes the proof of the Lipschitz continuity of g.

We will next show that g is differentiable. Indeed, let $x_0^{\prime }\in B_\delta ^{\prime }$. Now, the improvement of flatness at $x_0=(x_0^{\prime },g(x_0^{\prime }))$ implies that

$$\displaystyle \begin{aligned} -C|x-x_0|{}^{1+\gamma}\le (x-x_0)\cdot\nu_{x_0}\le C|x-x_0|{}^{1+\gamma}, \end{aligned}$$

for any x = (x′, g(x′)) with $x'\in B_\delta ^{\prime }$. For the sake of simplicity, we set $\nu :=\nu _{x_0}$ and $\nu =(\nu ',\nu _d)\in \mathbb {R}^{d-1}\times \mathbb {R}$. Since

$$\displaystyle \begin{aligned} (x-x_0)\cdot\nu_{x_0}=(x'-x_0^{\prime})\cdot \nu'+\left(g(x')-g(x_0^{\prime})\right)\nu_d, \end{aligned}$$

we get that

$$\displaystyle \begin{aligned} \left|g(x')-g(x_0^{\prime})-(x'-x_0^{\prime})\cdot \frac{\nu'}{\nu_d}\right|\le \frac{C}{\nu_d}(1+{\varepsilon})^{1+\gamma}|x'-x_0^{\prime}|{}^{1+\gamma}. \end{aligned}$$

This implies that g is differentiable at $x_0^{\prime }$ and that $\nabla g(x_0^{\prime })=\frac {\nu '}{\nu _d}$. Finally, the α-Hölder continuity of $\nabla g:B_\delta ^{\prime }\to \mathbb {R}^{d-1}$ follows by the γ-Hölder continuity of the map x↦ν _x. Indeed, for any $x',y'\in B_\delta ^{\prime }$, x = (x′, g(x′)) and y = (y′, g(y′)) we have that

$$\displaystyle \begin{aligned} |\nu_x-\nu_y|\le |x-y|{}^\alpha\le (1+{\varepsilon})^\alpha|x'-y'|{}^\alpha, \end{aligned}$$

which implies the Hölder continuity of all the components of the map $B_\delta ^{\prime }\ni x\mapsto \nu _x\in \mathbb {R}^d$ and thus, of the gradient ∇g. This concludes the proof of Proposition 8.6. □