Improvement of Flatness

Abstract

In this chapter, we will prove that the regular part of the free boundary Reg(∂ Ω_u) (defined in Sect. 6.4) is C ^{1, α} regular, for every α ∈ (0, 1∕2).

In this chapter, we will prove that the regular part of the free boundary Reg(∂ Ω_u) (defined in Sect. 6.4) is C ^{1, α} regular, for every α ∈ (0, 1∕2). We will first show that the minimizers of $\mathcal F_\Lambda$ are viscosity solutions of an overdetermined boundary value problem. Precisely, we will prove the following result.

FormalPara Proposition 7.1 (Local Minimizers Are Viscosity Solutions)

Let D be a bounded open set of $\mathbb {R}^d$ and let u ∈ H ¹(D) be a minimizer of $\mathcal F_\Lambda$ in D. Then, u is a viscosity solution of

$$\displaystyle \begin{aligned} \Delta u=0\quad \mathit{\text{in}}\quad \Omega_u,\qquad |\nabla u|=\sqrt{\Lambda} \quad \mathit{\text{on}}\quad \partial\Omega_u\cap D, \end{aligned} $$

(7.1)

in the sense of Definition 7.6.

The rest of the section is dedicated to the De Silva improvement of flatness theorem [23]. Precisely, we will prove that the (viscosity) solutions to (7.1) have C ^{1, α} regular free boundary. The proof follows step-by-step (sometimes with minor modifications) the original proof of De Silva [23].

Without loss of generality, we can assume that Λ = 1. This is due to the following remark, which is an immediate consequence of the definition of a viscosity solution (Definition 7.6).

FormalPara Remark 7.2

The continuous non-negative function $u:B_1\to \mathbb {R}$ is a viscosity solution to (7.1), for some Λ > 0, if and only if the function v := Λ^−1∕2 u is a viscosity solution to

$$\displaystyle \begin{aligned} \Delta v=0\quad \text{in}\quad \Omega_v,\qquad |\nabla v|=1 \quad \text{on}\quad \partial\Omega_v\cap D. \end{aligned} $$

(7.2)

As a consequence, it is sufficient to give the notion of flatness in the case Λ = 1.

FormalPara Definition 7.3 (Flatness)

Let $u:B_1\to \mathbb {R}$ be a given function. Let ε > 0 be a fixed real number and $\nu \in \mathbb {R}^{d}$ a unit vector. We say that

$$\displaystyle \begin{aligned} u\ \mathit{is}\ {\varepsilon}\mathit{-flat,}\ \mathit{in}\ \mathit{the}\ \mathit{direction}\ \nu,\ \mathit{in}\ B_{1}, \end{aligned}$$

if

$$\displaystyle \begin{aligned} {} \displaystyle \left(x\cdot \nu- {\varepsilon}\right)_{+}\le u(x)\le \left(x\cdot \nu+{\varepsilon}\right)_{+}\qquad \text{for every}\qquad x\in B_{1}. \end{aligned}$$

FormalPara Theorem 7.4 (Improvement of Flatness for Viscosity Solutions, De Silva [23])

There are dimensional constants C ₀ > 0, ε ₀ > 0, σ ∈ (0, 1) and r ₀ > 0 such that the following holds:

If $u:B_1\to \mathbb {R}$ be a continuous function such that:

• (a) u is non-negative and 0 ∈ ∂ Ω _u;

• (b) u is a viscosity solution to

  $$\displaystyle \begin{aligned} \begin{cases}\Delta u=0\quad \mathit{\text{in}}\quad \Omega_u\cap B_1\,,\\ |\nabla u|=1\quad \mathit{\text{on}}\quad \partial \Omega_u\cap B_1\,; \end{cases}\end{aligned}$$

• (c) there is ε ∈ (0, ε ₀] such that u is ε-flat in B ₁ , in the direction of the unit vector $\nu \in \mathbb {R}^d$.

Then, there is a unit vector $\tilde \nu \in \partial B_1\subset \mathbb {R}^d$ such that:

• (i) $\displaystyle |\tilde \nu -\nu |\le C_0 {\varepsilon }$;

• (ii) the function $u_{r_0}:B_1\to \mathbb {R}$ is σε-flat in B ₁ , in the direction $\tilde \nu$ , where we recall that $\displaystyle u_{r_0}(x)=\frac 1{r_0}u(r_0x)$.

Precisely, for any ε ₀ > 0, we may take

$$\displaystyle \begin{aligned} C_0=C_d,\qquad {\varepsilon}_0=r_0\qquad \mathit{\text{and}}\qquad \sigma= C_dr_0, \end{aligned}$$

where C _d is a dimensional constant.

From the improvement of flatness (Theorem 7.4) we will deduce the regularity of the free boundary (see Chap. 8). The section is organized as follows:

• In Sect. 7.1 we give the definition of a viscosity solution and we prove Proposition 7.1 using as competitors the radial solutions from the Propositions 2.15 and 2.16.

• In order to prove Theorem 7.4, we will reason by contradiction. This means that we will need a compactness result for a sequence of viscosity solutions $u_n:B_1\to \mathbb {R}$ which are ε _n-flat in B ₁ (ε _n being an infinitesimal sequence). This will be the aim of Sects. 7.2 and 7.3. In Sect. 7.2, we will prove the so-called Partial Harnack inequality (see Theorem 7.7), which we will use in Sect. 7.3 to obtain the compactness result (Lemma 7.14).

• Section 7.4 is dedicated to the proof of Theorem 7.4.

• Sections 8.1 and 8.2 are dedicated to the proof of Theorem 8.1, which is based on a classical argument and is well-known to be a consequence of the improvement of flatness Theorem 7.4.

7.1 The Optimality Condition on the Free Boundary

In this section, we give the definition of a viscosity solution and we prove Proposition 7.1.

Definition 7.5

Suppose that $\Omega \subset \mathbb {R}^d$ is an open set and that u is a continuous function, defined on the closure $\overline \Omega$. Let $x_0\in \overline \Omega$. We say that the function $\phi \in C^{\infty }(\mathbb {R}^d)$ touches u from below (resp. from above) at x ₀ in Ω if:

• u(x ₀) = ϕ(x ₀);

• there is a neighborhood $\mathcal N(x_0)\subset \mathbb {R}^d$ of x ₀ such that u(x) ≥ ϕ(x) (resp. u(x) ≤ ϕ(x)), for every $x\in \mathcal N(x_0)\cap \overline \Omega$.

Definition 7.6 (Viscosity Solutions)

Let $D\subset \mathbb {R}^d$ be an open set, A > 0 and $u:D\to \mathbb {R}^+$ be a continuous function. We say that u is a viscosity solution of the problem

$$\displaystyle \begin{aligned} {} \Delta u=0\quad \text{in}\quad \Omega_u,\qquad |\nabla u|=A \quad \text{on}\quad \partial\Omega_u\cap D, \end{aligned}$$

if for every $x_0\in \overline \Omega _u\cap D$ and ϕ ∈ C ^∞(D), we have

• if x ₀ ∈ Ω_u = {u > 0} and

  • if ϕ touches u from below at x ₀ in Ω_u, then Δϕ(x ₀) ≤ 0;

  • if ϕ touches u from above at x ₀ in Ω_u, then Δϕ(x ₀) ≥ 0;

• if x ₀ ∈ ∂ Ω_u ∩ D and

  • if ϕ touches u from below at x ₀ in Ω_u, then |∇ϕ(x ₀)|≤ A;

  • if ϕ ⁺ touches u from above at x ₀ in Ω_u, then |∇ϕ(x ₀)|≥ A.

Proof of Proposition 7.1

Suppose that x ₀ ∈ Ω_u and that ϕ ∈ C ^∞(D) touches u from below in x ₀. Since u is harmonic (and smooth) in the (open) set Ω_u, we get that Δϕ(x ₀) ≥ 0. The case when ϕ touches u from above at x ₀ ∈ Ω_u is analogous. Let now x ₀ ∈ ∂ Ω_u. Suppose that ϕ touches u from below at x ₀ and that |∇ϕ(x ₀)| > 1. We assume that x ₀ = 0 and we set

$$\displaystyle \begin{aligned} |\nabla \phi(0)|=a\qquad \text{and}\qquad \nu=\frac 1a\,\nabla \phi(0)\in \partial B _1\,, \end{aligned}$$

we get that, for some ρ > 0 small enough,

$$\displaystyle \begin{aligned} u(x)\ge \phi_+(x)\ge \frac{1+a}{2}\,(x\cdot\nu)_+\qquad \text{for every}\qquad x\in B_\rho. \end{aligned}$$

Let now r > 0 be large enough such that the radial solution u _r from Lemma 2.15 satisfies

Let $\widetilde u_{\varepsilon }$ be the following translation of u _r

$$\displaystyle \begin{aligned} \widetilde u_{\varepsilon}(x):=u_r\big(x-(R-{\varepsilon})\nu\big). \end{aligned}$$

Choosing ε small enough we can suppose that $\tilde u_{\varepsilon }(0)>0$ but

$$\displaystyle \begin{aligned} \widetilde u_{\varepsilon}(x)\le \frac{1+a}{2}\,(x\cdot\nu)_+\qquad \text{for every}\qquad x\in \partial B_\rho\,. \end{aligned}$$

Thus,

$$\displaystyle \begin{aligned} \widetilde u_{\varepsilon}\vee u=u \qquad \text{and}\qquad \widetilde u_{\varepsilon}\wedge u=\widetilde u_{\varepsilon} \quad \text{on}\quad \partial B_\rho\,. \end{aligned}$$

Now, since both $\widetilde u_{\varepsilon }$ and u are both minimizers in B _ρ, we get

$$\displaystyle \begin{aligned} \mathcal F_\Lambda(\widetilde u_{\varepsilon},B_\rho)\le \mathcal F_\Lambda( \widetilde u_{\varepsilon}\wedge u,B_\rho)\qquad \text{and}\qquad \mathcal F_\Lambda(u,B_\rho)\le \mathcal F_\Lambda( \widetilde u_{\varepsilon}\vee u,B_\rho). \end{aligned}$$

On the other hand, we have

$$\displaystyle \begin{aligned} \mathcal F_\Lambda(\widetilde u_{\varepsilon},B_\rho)+F_\Lambda(u,B_\rho)= \mathcal F_\Lambda( \widetilde u_{\varepsilon}\wedge u,B_\rho)+\mathcal F_\Lambda( \widetilde u_{\varepsilon}\vee u,B_\rho), \end{aligned}$$

which gives that

$$\displaystyle \begin{aligned} \mathcal F_\Lambda(\widetilde u_{\varepsilon},B_\rho)= \mathcal F_\Lambda( \widetilde u_{\varepsilon}\wedge u,B_\rho)\qquad \text{and}\qquad \mathcal F_\Lambda(u,B_\rho)= \mathcal F_\Lambda(\widetilde u_{\varepsilon}\vee u,B_\rho). \end{aligned}$$

Now, we define the function

and we set $v_r(x)=\widetilde v_{\varepsilon }\big (x+(R-{\varepsilon })\nu \big )$. Thus, we get that $\mathcal F_\Lambda (v_r,\mathbb {R}^d)=\mathcal F_\Lambda (u_r,\mathbb {R}^d)$, but v _r ≠ u _r, which is a contradiction with Lemma 2.15. The case when ϕ touches u from above is analogous and follows by Lemma 2.16. □

7.2 Partial Harnack Inequality

In this section we prove a weak version of Theorem 7.4. We will assume that u satisfies the conditions (a), (b) and (c) of Theorem 7.4, which means, in particular, that u is ε-flat in some direction ν:

$$\displaystyle \begin{aligned} (x\cdot\nu-{\varepsilon})_+\le u(x)\le (x\cdot\nu+{\varepsilon})_+\quad \text{for every}\quad x\in B_{1}. \end{aligned}$$

Then, we will prove that the flatness of u is improved in some smaller ball B _r. Precisely, we will show that

$$\displaystyle \begin{aligned} \big(x\cdot\nu-(1-c){\varepsilon}\big)_+\le u(x)\le \big(x\cdot\nu+(1-c){\varepsilon}\big)_+\quad \text{for every}\quad x\in B_{r}, \end{aligned} $$

(7.3)

for some dimensional constant c ∈ (0, 1). There are two main differences with respect to Theorem 7.4:

• The flatness might not really be improved in the sense of Theorem 7.4 and Fig. 7.1. Indeed, (7.3) only implies that the rescaled function

  $$\displaystyle \begin{aligned} u_r:B_1\to\mathbb{R},\quad u_r(x)=\frac 1ru(rx), \end{aligned}$$

  is $(1-c)\frac {{\varepsilon }}{r}$—flat in B ₁. Since the constants c and r are small, we might have

  $$\displaystyle \begin{aligned} (1-c)\frac{{\varepsilon}}{r}\ge{\varepsilon}, \end{aligned}$$

  which means that u _r might not be flatter than u.

  Fig. 7.1

  

  Improvement of flatness in the ball B ₁. For simplicity, we set r := r ₀

• The flatness direction does not change (ν′ = ν). Notice that, without changing the direction, the improvement of flatness (in the sense of Theorem 7.4) should not hold. In fact, the function $u(x)=x_d^+$ is ε-flat in the direction ν (whenever |ν − e _d| = ε), but for any r > 0, $u_r(x)=u(x)=x_d^+$, thus u _r cannot be more than ε-flat in the direction ν (the improvement is only possible if we are allowed to replace ν by a vector, which is closer to e _d).

The main result of this section is the following.

Theorem 7.7 (Partial Boundary Harnack)

There are dimensional constants $\bar {\varepsilon }>0$ and c ∈ (0, 1) such that for every viscosity solution u of (7.2) in $B_1\subset \mathbb {R}^d$ such that $0\in \overline \Omega _u$ we have the following property:

If there are two real numbers a ₀ < b ₀ such that

$$\displaystyle \begin{aligned} |b_0-a_0|\le \bar{\varepsilon} \qquad \mathit{\text{and}}\qquad (x_d+a_0)_+\le u(x)\le (x_d+b_0)_+\quad \mathit{\text{on}}\quad B_1, \end{aligned}$$

then there are real numbers a ₁ and b ₁ such that a ₀ ≤ a ₁ < b ₁ ≤ b ₀,

$$\displaystyle \begin{aligned} |b_1-a_1|\le (1-c)|a_0-b_0| \qquad \mathit{\text{and}}\qquad (x_d+a_1)_+\le u(x)\le (x_d+b_1)_+\quad \mathit{\text{on}}\quad B_{\frac1{20}}. \end{aligned}$$

Proof

Since $0\in \overline \Omega _u$ we have that a ₀ ≥−1∕10. We consider two cases:

• 1. Suppose that |a ₀|≤ 1∕10. Then applying Lemma 7.10 we have the claim.

• 2. Suppose that a ₀ ≥ 1∕10. Then u is harmonic in B ₁ ∩{x _d > −1∕10} (and so, in the ball B _1∕10) and the claim follows by Lemma 7.9.

□

We next prove the two main results: Lemmas 7.9 and 7.10. Section 7.2.1 is dedicated to the proof of Lemma 7.9, which is a consequence of the classical Harnack inequality for harmonic functions stated in Lemma 7.8. Section 7.2.2 is dedicated to the boundary version of the Harnack inequality (Lemma 7.10), which is due to De Silva [23].

7.2.1 Interior Harnack Inequality

Lemma 7.8 (Harnack Inequality)

There is a dimensional constant $C_{\mathcal H}$ such that for every $h:B_{2r}(x_0)\to \mathbb {R}$ , a non-negative harmonic function in the ball $B_{2r}(x_0)\subset \mathbb {R}^d$ , the following (Harnack) inequality does hold

$$\displaystyle \begin{aligned} \max_{B_r(x_0)} h\le C_{\mathcal H}\min_{B_r(x_0)} h\,. \end{aligned} $$

(7.4)

In particular, we have

$$\displaystyle \begin{aligned} {} h(x_0)\le C_{\mathcal H}\min_{B_r(x_0)} h\,. \end{aligned}$$

Proof

The proof is an immediate consequence of the mean-value property. □

Lemma 7.9 (Improvement of Flatness at Fixed Scale)

Let $C_{\mathcal H}>1$ be the dimensional constant from the Harnack inequality (7.4) and let $c_{\mathcal H}:=\big (2C_{\mathcal H}\big )^{-1}$ . Suppose that $u:B_{2r}\to \mathbb {R}$ is a harmonic function for which there are a constant ε > 0 and a linear function $\ell :\mathbb {R}^d\to \mathbb {R}$ such that

$$\displaystyle \begin{aligned} \ell(x)\le u(x)\le \ell(x)+{\varepsilon} \qquad \mathit{\text{for every}}\qquad x\in B_{2r}. \end{aligned}$$

Then at least one of the following does hold :

• (i) $\ell (x)+c_{\mathcal H}{\varepsilon }\le u(x)\le \ell (x)+{\varepsilon }\quad \mathit{\text{for every}}\quad x\in B_{r}$;

• (ii) $\ell (x)\le u(x)\le \ell (x)+(1-c_{\mathcal H}){\varepsilon }\quad \mathit{\text{for every}}\quad x\in B_{r}$.

Proof

We consider two cases.

Case 1. Suppose that u(0) ≥ ℓ(0) + ε∕2. Then the function h := u − ℓ is harmonic and non-negative in B _2r. Then, by the Harnack inequality (7.4), we have

$$\displaystyle \begin{aligned} \frac{\varepsilon}2\le h(0)\le C_{\mathcal H}\min_{B_r}h, \end{aligned}$$

which means that

$$\displaystyle \begin{aligned} u- \ell\ge \frac{{\varepsilon}}{2C_{\mathcal H}}\quad \text{in}\quad B_r, \end{aligned}$$

and so (i) holds.

Case 2. Suppose that u(0) ≤ ℓ(0) + ε∕2. Then the function h := ℓ + ε − u is harmonic and non-negative in B _2r. Then, by the Harnack inequality (7.4), we have

$$\displaystyle \begin{aligned} \frac{\varepsilon}2\le h(0)\le C_{\mathcal H}\min_{B_r}h, \end{aligned}$$

which means that

$$\displaystyle \begin{aligned} \ell+{\varepsilon}-u\ge \frac{{\varepsilon}}{2C_{\mathcal H}}\quad \text{in}\quad B_r, \end{aligned}$$

and so (ii) holds. □

7.2.2 Partial Harnack Inequality at the Free Boundary

Lemma 7.10 (Improving the Flatness at Fixed Scale; De Silva [23])

There are dimensional constants $\bar {\varepsilon }>0$ and c ∈ (0, 1), for which the following does hold.

Suppose that $u:B_1\to \mathbb {R}$ is a continuous non-negative function and a viscosity solution of (7.2) in $B_1\subset \mathbb {R}^d$ . Then, we have the following property:

If there are real constants ε and σ, $0<{\varepsilon }\le \bar {\varepsilon }$ and |σ| < 1∕10, such that

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

then at least one of the following does hold :

• (i) (x _d + σ + cε)₊ ≤ u(x) ≤ (x _d + σ + ε)₊ for every x ∈ B _1∕20 ;

• (ii) (x _d + σ)₊ ≤ u(x) ≤ (x _d + σ + (1 − c)ε)₊ for every x ∈ B _1∕20.

Proof

We set

$$\displaystyle \begin{aligned} \displaystyle\bar x=\frac{e_d}5\qquad \text{and}\qquad \bar c=\left(20^d-\big({4}/{3}\big)^d\right)^{-1}\ , \end{aligned}$$

and consider the function $w:\mathbb {R}^d\to \mathbb {R}$, defined as (see Fig. 7.2):

Fig. 7.2

The function w

The set, where the function w is not constantly vanishing, is precisely the ball $B_{{3}/{4}}(\bar x)$ (see Fig. 7.2). Moreover, on the annulus , the function w has the following properties :

• (w1) $\displaystyle \Delta w(x)=2\,d\,\bar c \,|x-\bar x|{ }^{-(d+2)}\ge 2\,d\,\bar c\, \big ({4}/{3}\big )^{d+2}>0\$.

• (w2) $\displaystyle \partial _{x_d}w\ge C_{w}>0$ on the half-space {x _d < 1∕10}. Here, C _w > 0 is an (explicit) constant depending only on the dimension.

Following the notation from [23] we set p(x) = x _d + σ. Similarly to what we did in the proof of Lemma 7.9, we consider two cases.

Case 1. Suppose that $u(\bar x)\ge p(\bar x)+\frac {{\varepsilon }}2$.

Since the function u − p is harmonic and non-negative in the ball $B_{{1}/{10}}(\bar x)$, we can apply the Harnack inequality (7.4). Thus, setting $c_{\mathcal H}:=\big (2C_{\mathcal H}\big )^{-1}$ we get

$$\displaystyle \begin{aligned} u(x)-p(x)\ge c_{\mathcal H}{\varepsilon}\quad \text{in}\quad B_{{1}/{20}}(\bar x). \end{aligned}$$

We now consider the family of functions

$$\displaystyle \begin{aligned} v_t(x)=p(x)+c_{\mathcal H}{\varepsilon} w(x)-c_{\mathcal H}{\varepsilon}+c_{\mathcal H}{\varepsilon} t. \end{aligned}$$

We will prove that for every t ∈ [0, 1), we have u(x) ≥ v _t(x) in B ₁. We notice that, for t < 1 the function v _t has the following properties:

• (v1) v _t(x) < p(x) ≤ u(x) on (since the support of w is precisely $B_{{3}/{4}}(\bar x)$),

• (v2) v _t(x) < u(x) in $\overline B_{\frac 1{20}}(\bar x)$ (by the choice of the constant $c_{\mathcal H}$),

• (v3) Δv _t(x) > 0 on the blue annulus (follows from (w1)),

• (v4) $|\nabla v_t|(x)\ge \partial _{x_d}v_t(x)\ge 1+ c_{\mathcal H}{\varepsilon } C_w>1$ on .

Suppose (by absurd) that, for some t ∈ [0, 1), the function u − v _t has local minimum in B ₁ in a point $x\in \overline B_1$. By (v3) and the fact that u is a viscosity solution we have that . By (v4) we have that and . Thus we get . By (v1) and (v2) we conclude that

$$\displaystyle \begin{aligned} \min_{x\in \overline B_1} \big\{u(x)-v_t(x)\big\}>0\qquad \text{whenever}\qquad t<1. \end{aligned}$$

Thus, we obtain that u ≥ v ₁ on $\overline B_1$, i.e.

$$\displaystyle \begin{aligned} u(x)\ge p(x)+c_{\mathcal H}{\varepsilon} w(x)\quad \text{on}\quad B_1. \end{aligned}$$

Now since w is strictly positive on the ball B _1∕20 we get that

$$\displaystyle \begin{aligned} u(x)\ge p(x)+c_{d}{\varepsilon} \quad \text{on}\quad B_{\frac1{20}}, \end{aligned}$$

which proves that the property (i) holds.

Case 2. Suppose that $u(\bar x)< p(\bar x)+\frac {{\varepsilon }}2$.

Since the function p + ε − u is harmonic and non-negative in the ball $B_{{1}/{10}}(\bar x)$, we can apply the Harnack inequality, thus obtaining that for a dimensional constant $c_{\mathcal H}>0$ we have

$$\displaystyle \begin{aligned} p+{\varepsilon}-u\ge c_{\mathcal H}{\varepsilon}\quad \text{in}\quad B_{{1}/{20}}(\bar x). \end{aligned}$$

We now consider the family of functions

$$\displaystyle \begin{aligned} v_t(x)=p(x)+{\varepsilon}-c_{\mathcal H}{\varepsilon} w(x)+c_{\mathcal H}{\varepsilon}-c_{\mathcal H}t, \end{aligned}$$

and, reasoning as in the previous case, we get that

$$\displaystyle \begin{aligned} v_t(x)^+\ge u(x)\qquad \text{for every}\qquad t\in[0,1). \end{aligned}$$

In particular, since w is strictly positive on the ball B _1∕20, we get that

$$\displaystyle \begin{aligned} u(x)\le \big(p(x)+(1-c_d){\varepsilon}\big)_+ \quad \text{on}\quad B_{{1}/{20}}, \end{aligned}$$

which concludes the proof. □

7.3 Convergence of Flat Solutions

In this subsection we prove the compactness result that we will need in the proof of Theorem 7.4. The proof is entirely based on Theorem 7.7, from which we know that any (continuous, non-negative) viscosity solution $u:B_1\to \mathbb {R}$ of (7.2) satisfies the following condition.

Condition 7.11 (Partial Improvement of Flatness)

There are constants $\bar {\varepsilon }>0$ and c ∈ (0, 1) such that the following holds. If $x_0\in \overline \Omega _u$ , B _r(x ₀) ⊂ B ₁ and a ₀ < b ₀ are such that

$$\displaystyle \begin{aligned} |b_0-a_0|\le r{\bar{\varepsilon}} \qquad \mathit{\text{and}}\qquad (x_d+a_0)_+\le u(x)\le (x_d+b_0)_+\quad \mathit{\text{on}}\quad B_r(x_0), \end{aligned}$$

then there are real numbers a ₁ and b ₁ such that a ₀ ≤ a ₁ < b ₁ ≤ b ₀,

$$\displaystyle \begin{aligned} |b_1-a_1|\le (1-c)|a_0-b_0| \qquad \mathit{\text{and}}\qquad (x_d+a_1)_+\le u(x)\le (x_d+b_1)_+\quad \mathit{\text{on}}\quad B_{{r}/{20}}(x_0). \end{aligned}$$

Remark 7.12

We notice that if $u:B_1\to \mathbb {R}$ is a continuous non-negative function on B ₁, then, for any a < b and any set E ⊂ B ₁, the inequality

$$\displaystyle \begin{aligned} (x_d+a)_+\le u(x)\le (x_d+b)_+\quad \text{on}\quad E, \end{aligned}$$

is equivalent to

$$\displaystyle \begin{aligned} x_d+a \le u(x)\le x_d+b\quad \text{on}\quad E\cap \overline\Omega_u, \end{aligned}$$

Thus, an equivalent way to state Condition 7.11 is the following. The non-negative function $u:B_1\to \mathbb {R}$ satisfies Condition 7.11, if and only if, the following holds.

If $x_0\in \overline \Omega _u$, B _r(x ₀) ⊂ B ₁ and a ₀ < b ₀ are such that

$$\displaystyle \begin{aligned} |b_0-a_0|\le r{\bar{\varepsilon}} \qquad \text{and}\qquad x_d+a_0\le u(x)\le x_d+b_0\quad \text{on}\quad B_r(x_0)\cap\overline\Omega_u, \end{aligned}$$

then there are real numbers a ₁ and b ₁ such that a ₀ ≤ a ₁ < b ₁ ≤ b ₀,

$$\displaystyle \begin{aligned} |b_1-a_1|\le (1-c)|a_0-b_0| \qquad \text{and}\qquad x_d+a_1\le u(x)\le x_d+b_1\quad \text{on}\quad B_{{r}/{20}}(x_0)\cap\overline\Omega_u. \end{aligned}$$

The constants $\bar {\varepsilon }$ and c are the same as in Condition 7.11.

Lemma 7.13

Suppose that the continuous non-negative function $u:B_1\to \mathbb {R}$ satisfies Condition 7.11 with constants c and $\bar {\varepsilon }$ . Suppose that $0\in \overline \Omega _u$ and that there are two real numbers a ₀ < b ₀ such that

$$\displaystyle \begin{aligned} {\varepsilon}:=|b_0-a_0|< \frac{\bar{\varepsilon}}2 \qquad \mathit{\text{and}}\qquad x_d+a_0\le u(x)\le x_d+b_0\quad \mathit{\text{on}}\quad B_1\cap\overline\Omega_u. \end{aligned}$$

Then, setting

$$\displaystyle \begin{aligned} \tilde u(x)=\frac{u(x)-x_d}{{\varepsilon}}\qquad \mathit{\text{for every}}\qquad x\in \overline\Omega_u\cap B_1, \end{aligned}$$

for every $x_0\in B_{{1}/{2}}\cap \overline \Omega _{u}$ , we have the uniform estimate

where C is a numerical constant and γ depends only on c.

Proof

Let n ≥ 0 be such that

$$\displaystyle \begin{aligned} \frac 12\big({1}/{20}\big)^{n+1}\le\,\frac{{\varepsilon}}{\bar{\varepsilon}}\,< \frac 12\big({1}/{20}\big)^{n}. \end{aligned}$$

Let $\displaystyle r_j=\frac 12\big ({1}/{20}\big )^{j}.$ Then, we have

$$\displaystyle \begin{aligned} {\varepsilon}\le \bar {\varepsilon} r_j\qquad \text{for every}\qquad j=0,1,\dots,n. \end{aligned}$$

Thus, for every $x_0\in B_{1/2}\cap \overline \Omega _u$ we can apply the (partial) improvement of flatness in $B_{r_j}(x_0)$, for every j = 0, 1, …, n. Thus, we get that there are

$$\displaystyle \begin{aligned} a_0\le a_1\le \dots \le a_j\le \dots\le a_n\le b_n\le \dots\le b_j\le \dots\le b_1\le b_0 \end{aligned}$$

such that

$$\displaystyle \begin{aligned} |b_j-a_j|\le (1-c)^j|a_0-b_0| \qquad \text{and}\qquad (x_d+a_j)_+\le u(x)\le (x_d+b_j)_+\quad \text{on}\quad B_{r_j}(x_0), \end{aligned}$$

which implies that

$$\displaystyle \begin{aligned} x_d+a_j\le u(x)\le x_d+b_j\quad \text{on}\quad B_{r_j}(x_0)\cap\overline\Omega_u, \end{aligned}$$

and so,

$$\displaystyle \begin{aligned} \left|\big(u(x)-x_d\big)-a_j\right|\le (1-c)^j{\varepsilon} \quad \text{for}\quad x\in B_{r_j}(x_0)\cap\overline\Omega_u. \end{aligned}$$

The triangular inequality implies that

$$\displaystyle \begin{aligned} \left|\tilde u(x)-\tilde u(x_0)\right|\le 2(1-c)^j \quad \text{for every}\quad x\in B_{r_j}(x_0)\cap\overline\Omega_u, \end{aligned}$$

which gives the claim by choosing j such that

$$\displaystyle \begin{aligned} r_{j+1}<|x-x_0|\le r_j, \end{aligned}$$

and setting γ to be such that (1∕20)^γ = 1 − c. □

Lemma 7.14 (Compactness for Flat Sequences)

Let $\bar {\varepsilon }>0$ and c ∈ (0, 1) be fixed constants. Suppose that $u_k: B_1\to \mathbb {R}$ is a sequence of continuous non-negative functions such that

• (a) u _k satisfies Condition 7.11 in B ₁ with constants $\bar {\varepsilon }$ and c.

• (b) u _k is ε _k -flat in B ₁ , that is,

  $$\displaystyle \begin{aligned} x_d-{\varepsilon}_k\le u_k(x)\le x_d+{\varepsilon}_k\quad \mathit{\text{in}}\quad B_1\cap\overline\Omega_{u_k}. \end{aligned}$$

• (c) lim_k→∞ ε _k = 0.

Then there is a Hölder continuous function $\widetilde u : B_{{1}/{2}}\cap \{x_d\ge 0\}\to \mathbb {R}$ and a subsequence of

$$\displaystyle \begin{aligned} \displaystyle \tilde u_k(x)=\frac{u_k(x)-x_d}{{\varepsilon}_k}\ ,\qquad u_k: B_{{1}/{2}}\cap\overline\Omega_{u_k}\to \mathbb{R}, \end{aligned}$$

that we still denote by $\tilde u_k$ such that the following claims do hold.

• (i) For every δ > 0, $\tilde u_k$ converges uniformly to $\widetilde u$ on the set B _1∕2 ∩{x _d ≥ δ}.

• (ii) The sequence of graphs

  $$\displaystyle \begin{aligned} \Gamma_k=\big\{(x,\tilde u_k(x))\ :\ x\in \overline\Omega_{u_k}\cap B_{{1}/{2}}\big\}\subset \mathbb{R}^{d+1}, \end{aligned}$$

  converges in the Hausdorff distance (in $\mathbb {R}^{d+1}$ ) to the graph

  $$\displaystyle \begin{aligned} \Gamma=\big\{(x,\tilde u(x))\ :\ x\in B_{{1}/{2}}\cap \{x_d\ge 0\}\big\}. \end{aligned}$$

Proof

We first prove (i). For every $y\in B_{{1}/{2}}\cap \overline \Omega _{u_k}$ we have that

$$\displaystyle \begin{aligned} x_d-{\varepsilon}_k\le u_k(x)\le x_d+{\varepsilon}_k\quad \text{for every}\quad x\in B_{{1}/{2}}(y)\cap \overline \Omega_{u_k}. \end{aligned}$$

Thus, by Lemma 7.13 we have that $\tilde u_k$ satisfies

$$\displaystyle \begin{aligned} |\tilde u_k(x)-\tilde u_k(y)|\le C|x-y|{}^\gamma\quad \text{for every}\quad x\in B_{{1}/{2}}(y)\cap \overline \Omega_{u_k}\quad \text{such that}\quad |x-y|\ge \frac{{\varepsilon}_k}{\bar{\varepsilon}}, \end{aligned}$$

which, since y is arbitrary, gives

$$\displaystyle \begin{aligned} |\tilde u_k(x)-\tilde u_k(y)|\le C|x-y|{}^\gamma\quad \text{for every}\quad x,y\in B_{{1}/{2}}\cap \overline \Omega_{u_k}\quad \text{such that}\quad |x-y|\ge \frac{{\varepsilon}_k}{\bar{\varepsilon}}. \end{aligned}$$

Since, for ε _k ≤ δ, we have that $\{x_d\ge \delta \}\cap B_{1}\subset \Omega _{u_k}\cap B_1$, we get that the sequence $\tilde u_k : \{x_d\ge \delta \}\cap B_{{1}/{2}}\to \mathbb {R}$ satisfies :

• $\tilde u_k$ is equi-bounded on {x _d ≥ δ}∩ B _1∕2

  $$\displaystyle \begin{aligned} -1=\frac{(x_d-{\varepsilon}_k)-x_d}{{\varepsilon}_k}\le \frac{u_k(x)-x_d}{{\varepsilon}_k}\le\frac{(x_d+{\varepsilon}_k)-x_d}{{\varepsilon}_k}=1 ; \end{aligned}$$

• $\tilde u_k$ satisfies

  $$\displaystyle \begin{aligned} \text{osc}\,\big(\tilde u_k ; {A_{2r,r}(x_0)\cap \{x_d\ge \delta\}\cap B_{{1}/{2}}}\big)\le 2Cr^\gamma\qquad \text{for every}\qquad r\ge \frac{{\varepsilon}_k}{\bar{\varepsilon}}, \end{aligned}$$

  where, for any set $E\subset \overline \Omega _{u_k}$, we define:

  $$\displaystyle \begin{aligned} \text{osc}\,(\tilde u_k ; E):=\sup_{E} \tilde u_k-\inf_{E}\tilde u_k, \end{aligned}$$

  and, for every 0 < r < R, A _R,r(x ₀) is the annulus

  

Thus, by the Ascoli-Arzelà Theorem, there is a subsequence converging uniformly on the set {x _d ≥ δ}∩ B _1∕2 to a Holder continuous function

$$\displaystyle \begin{aligned} \tilde u\ :\ \{x_d\ge \delta\}\cap B_{{1}/{2}}\to [-1,1], \end{aligned}$$

satisfying

$$\displaystyle \begin{aligned} |\tilde u(x)-\tilde u(y)|\le C|x-y|{}^\gamma\qquad \text{for every}\qquad x,y\in B_{{1}/{2}}\cap \{x_d\ge \delta\}. \end{aligned}$$

The above argument does not depend on δ > 0. Thus, the function $\tilde u$ can be defined on the entire half-ball {x _d > 0}∩ B _1∕2. Moreover, the constants C and γ do not depend on the choice of δ > 0. This implies that we can extend $\tilde u$ to a Hölder continuous function

$$\displaystyle \begin{aligned} \tilde u\ :\ \{x_d\ge 0\}\cap B_{{1}/{2}}\to [-1,1]. \end{aligned}$$

still satisfying the uniform continuity estimate

$$\displaystyle \begin{aligned} |\tilde u(x)-\tilde u(y)|\le C|x-y|{}^\gamma\qquad \text{for every}\qquad x,y\in B_{{1}/{2}}\cap \{x_d\ge 0\}. \end{aligned}$$

We now prove (ii). Suppose that $\tilde x=(x,\tilde u(x))\in \Gamma$. For every δ > 0, there is a point y ∈ B _1∕2 ∩{x _d > δ∕2} such that |x − y|≤ δ. (Notice that, if x ∈ B _1∕2 ∩{x _d > δ∕2}, then we can simply take y = x.) Then, setting $\tilde y=(y,\tilde u(y))$, we have the estimate

$$\displaystyle \begin{aligned} |\tilde x-\tilde y|{}^2=|x-y|{}^2+|\tilde u(x)-\tilde u(y)|{}^2\le\delta^2+C^2\delta^{2\gamma}. \end{aligned}$$

On the other hand, for every k such that ε _k ≤ δ, we have

$$\displaystyle \begin{aligned} \text{dist}\big(\tilde y,\Gamma_k\big)\le |\tilde u(y)-\tilde u_k(y)|\le \|\tilde u-\tilde u_k\|{}_{L^\infty(B_{{1}/{2}}\cap\{x_d>\delta/2\})}. \end{aligned}$$

Thus, we finally obtain the estimate

$$\displaystyle \begin{aligned} {} \text{dist}\big(\tilde x,\Gamma_k\big)\le \big(\delta^2+C^2\delta^{2\gamma}\big)^{{1}/{2}}+\|\tilde u-\tilde u_k\|{}_{L^\infty(B_{{1}/{2}}\cap\{x_d>\delta/2\})}. \end{aligned}$$

Let now $\tilde x_k=\big (x_k,\tilde u_k(x_k)\big )\in \Gamma _k$. Let k be such that ${{\varepsilon }_k}/{\bar {\varepsilon }}\le \frac {\delta }2$. Let y _k ∈{x _d ≥ δ}∩ B _1∕2 be such that δ∕2 ≤|x _k − y _k|≤ 2δ and let $\tilde y_k=\big (y_k,\tilde u_k(y_k)\big )$. Then, we have

$$\displaystyle \begin{aligned} |\tilde x_k-\tilde y_k|{}^2=|x_k-y_k|{}^2+|\tilde u_k(x_k)-\tilde u_k(y_k)|{}^2\le4\delta^2+4C^2\delta^{2\gamma}. \end{aligned}$$

Reasoning as above, we get

$$\displaystyle \begin{aligned} {} \text{dist}\big(\tilde x_k,\Gamma\big)\le 2\big(\delta^2+C^2\delta^{2\gamma}\big)^{{1}/{2}}+\|\tilde u-\tilde u_k\|{}_{L^\infty(B_{{1}/{2}}\cap\{x_d>\delta\})}. \end{aligned}$$

Now, since δ is arbitrary and $\tilde u_k$ converges to $\tilde u$ uniformly on {x _d > δ∕2}∩ B _1∕2, we get that

$$\displaystyle \begin{aligned} \lim_{k\to\infty}\text{dist}_{\mathcal H}(\Gamma_k,\Gamma)=0. \end{aligned}$$

□

7.4 Improvement of Flatness: Proof of Theorem 7.4

In this subsection, we prove Theorem 7.4. Since, we will reason by contradiction, we will first study the limits of the sequences of (flat) viscosity solutions to (7.2) in B ₁.

Lemma 7.15 (The Linearized Problem)

Suppose that $u_k:B_1\to \mathbb {R}$ is a sequence of continuous non-negative functions such that:

• (a) for every k, u _k is a viscosity solution of

  $$\displaystyle \begin{aligned} \Delta u_k=0\quad \mathit{\text{in}}\quad \Omega_{u_k}\cap B_1\ ,\qquad |\nabla u_k|=1\quad \mathit{\text{on}}\quad \partial \Omega_{u_k}\cap B_1. \end{aligned} $$

  (7.5)

• (b) for every k, u _k is ε _k -flat in B ₁ in the sense that

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

• (c) lim_k→∞ ε _k = 0.

Then, up to extracting a subsequence, the sequence of functions

$$\displaystyle \begin{aligned} \displaystyle \widetilde u_k\,:\,B_{{1}/{2}}\cap\overline\Omega_{u_k}\to \mathbb{R}\ ,\qquad \widetilde u_k(x)=\frac{u_k(x)-x_d}{{\varepsilon}_k}\,, \end{aligned}$$

converges (in the sense of Lemma 7.14 (i) and (ii)) to a Hölder continuous function

$$\displaystyle \begin{aligned} \widetilde u : B_{{1}/{2}}\cap \{x_d\ge 0\}\to\mathbb{R}. \end{aligned}$$

Moreover, we have that

• (i) $\tilde u$ is a viscosity solution to

  $$\displaystyle \begin{aligned} \Delta \tilde u=0\quad \mathit{\text{in}}\quad B_{{1}/{2}}\cap \{x_d>0\}\ ,\qquad \frac{\partial \tilde u}{\partial x_d}=0\quad \mathit{\text{on}}\quad B_{{1}/{2}}\cap \{x_d=0\}\ , \end{aligned} $$

  (7.6)

  in the sense that

  • $\tilde u$ is harmonic in B _1∕2 ∩{x _d > 0},

  • If P is a polynomial touching $\tilde u$ from below (above) in a point x ₀ ∈ B _1∕2 ∩{x _d = 0}, then $\displaystyle \frac {\partial P}{\partial x_d}(x_0)\le 0$ $\ \left (\displaystyle \frac {\partial P}{\partial x_d} (x_0)\ge 0\right )$.

• (ii) $\tilde u\in C^{\infty }\big (B_{{1}/{2}}\cap \{x_d\ge 0\}\big )$ and is a classical solution of (7.6).

Proof

The existence of the limit function $\tilde u$ follows by Lemma 7.14.

We first prove (i). Suppose that P is a polynomial touching $\tilde u$ strictly from below in a point x ₀ ∈ B _1∕2 ∩{x _d ≥ 0}. Then there exists a sequence of points $x_k\in \overline \Omega _{u_k}$ such that P touches $\tilde u_k$ from below in x _k and x _k → x ₀ as k →∞. We consider two cases:

• (1) Suppose that x ₀ ∈{x _d > 0}. Then there is some δ > 0 such that x _k ∈{x _d > δ}, for every k large enough. Thus, $x_k\in \Omega _{u_k}$ for k large enough and so, since $\tilde u_k$ is harmonic in $\Omega _{u_k}$, ΔP(x _k) ≥ 0. Passing to the limit as k →∞ we get ΔP(x ₀) ≥ 0.

• (2) Suppose that x ₀ ∈{x _d = 0}. We suppose without loss of generality that x ₀ = 0. We consider the family of polynomials

  $$\displaystyle \begin{aligned} P_{{\varepsilon}} (x)=P(x)+ \frac 1{\varepsilon} x_d^2-{\varepsilon} x_d. \end{aligned}$$

  In a sufficiently small neighborhood of zero, we have that P _ε still touches $\tilde u$ (strictly) from below in 0. Moreover,

  $$\displaystyle \begin{aligned} \Delta P_{\varepsilon}>0\quad \text{in a neighborhood of zero}\,,\qquad \frac{\partial P_{\varepsilon}}{\partial x_d}(0)=\frac{\partial P}{\partial x_d}(0)-{\varepsilon}. \end{aligned}$$

  Thus it is sufficient to show that for every ε > 0, we have $\displaystyle \frac {\partial P_{\varepsilon }}{\partial x_d}(0)\le 0$. Let now ε > 0 be fixed. Consider the sequence of points $x_k\in \overline \Omega _{u_k}$ such that P _ε touches $\tilde u_k$ from below in x _k and x _k → x ₀ as k →∞. Since ΔP _ε(x _k) > 0 and $\tilde u_k$ is harmonic in $\Omega _{u_k}$ we have that necessarily x _k ∈ ∂ Ω_k. By the definition of $\tilde u_k=\frac {u_k-x_d}{{\varepsilon }_k}$ we get that the polynomial Q(x) = ε _k P _ε(x) + x _d touches u _k from below in x _k. Since u _k is a viscosity solution of (7.5), we get that

  $$\displaystyle \begin{aligned} 1\ge |\nabla Q(x_k)|{}^2\ge \Big(1+{\varepsilon}_k\frac{\partial P_{\varepsilon}}{\partial x_d}(x_k)\Big)^2= 1+2{\varepsilon}_k\frac{\partial P_{\varepsilon}}{\partial x_d}(x_k)+{\varepsilon}_k^2\left|\frac{\partial P_{\varepsilon}}{\partial x_d}(x_k)\right|{}^2. \end{aligned}$$

  Thus, we have $\displaystyle \frac {\partial P_{\varepsilon }}{\partial x_d}(0)\le 0$, which concludes the proof after letting ε → 1.

We now prove (ii). We write $\mathbb {R}^d\ni x=(x',x_d)$ with $x'\in \mathbb {R}^{d-1}$ and $x_d\in \mathbb {R}$. We consider the function $w:\mathbb {R}^d\to \mathbb {R}$ defined by $\tilde u$ and its reflexion:

$$\displaystyle \begin{aligned} {} w(x',x_d)=\begin{cases}\tilde u(x',x_d),\ \text{ if }\ x_d\ge 0,\\ \tilde u(x',-x_d),\ \text{ if }\ x_d\le 0. \end{cases} \end{aligned}$$

We will prove that w is harmonic on $\mathbb {R}^d$. Suppose that P is a polynomial touching w strictly from below in a point x ₀ ∈{x _d = 0}. Since w is harmonic on {x _d ≠ 0} it is sufficient to prove that ΔP(x ₀) ≤ 0. We first notice that since w(x′, x _d) = w(x′, −x _d) then also the polynomial P(x′, −x _d) touches w strictly from below in x ₀ and, as a consequence, so does the polynomial

$$\displaystyle \begin{aligned} Q(x',x_d)=\frac{P(x',x_d)+P(x',-x_d)}{2}, \end{aligned}$$

which satisfies

$$\displaystyle \begin{aligned} \Delta Q=\Delta P\quad \text{and}\quad \frac{\partial Q}{\partial x_d}=0\quad \text{on}\quad \{x_d=0\}. \end{aligned}$$

Consider the polynomial

$$\displaystyle \begin{aligned} Q_{\varepsilon} (x)=Q(x)+{\varepsilon}\, x\cdot e_d. \end{aligned}$$

Then Q _ε touches w from below in a point x _ε and we have that x _ε → x ₀ as ε → 0. We notice that necessarily x _ε ∈{x _d ≥ 0}. Moreover, we can rule out the case x _ε ∈{x _d = 0} since by the hypothesis on $\tilde u$ we have that in this case we should have

$$\displaystyle \begin{aligned} 0\ge\frac{\partial Q_{\varepsilon}}{\partial x_d}(x_{\varepsilon})=\frac{\partial Q}{\partial x_d}(x_{\varepsilon})+{\varepsilon}={\varepsilon}, \end{aligned}$$

which is impossible. Thus x _ε ∈{x _d > 0} and since $\tilde u$ is harmonic in {x _d > 0} we get that

$$\displaystyle \begin{aligned} 0\ge\Delta Q_{\varepsilon}(x_{\varepsilon})=\Delta Q(x_{\varepsilon}). \end{aligned}$$

Passing to the limit as ε → 0, we obtain that ΔQ(x ₀) ≤ 0, which concludes the proof. □

Lemma 7.16 (First and Second Order Estimates for Harmonic Functions)

Suppose that $h:B_R\to \mathbb {R}$ is a bounded harmonic function in B _R . Then

$$\displaystyle \begin{aligned} \|\nabla h\|{}_{L^\infty(B_{\frac{R}2})}\le \frac{C_d}{R}\|h\|{}_{L^\infty{(B_R)}}\ , \end{aligned} $$

(7.7)

and

$$\displaystyle \begin{aligned} \big|h(x)-h(0)-x\cdot \nabla h(0)\big|\le \frac{C_d}{R^2}|x|{}^2\|h\|{}_{L^\infty{(B_R)}}\qquad \mathit{\text{for every}}\qquad x\in B_{{R}/{2}}\ , \end{aligned} $$

(7.8)

where C _d is a dimensional constant.

Proof

Let x ₀ ∈ B _3R∕4. Since h is harmonic in B _R∕4(x ₀), we have that also ∂ _i h is harmonic in the same ball B _R∕4(x ₀), we have

$$\displaystyle \begin{aligned} \partial_i h(x_0)=\frac{4^d}{\omega_dR^d}\int_{B_{\frac{R}4}(x_0)}\partial_i h(x)\,dx=\frac{4^d}{\omega_dR^d}\int_{B_{\frac{R}4}(x_0)}\text{div}\, X\,dx, \end{aligned}$$

where X = (0, …, h, …, 0) is the vector with the only non-zero component being the ith one, which is precisely h. Now, the divergence theorem gives

$$\displaystyle \begin{aligned} \partial_i h(x_0)=\frac{4^d}{\omega_dR^d}\int_{\partial B_{\frac{R}4}(x_0)}\nu\cdot X\,d\mathcal{H}^{d-1}=\frac{4^d}{\omega_dR^d}\int_{\partial B_{\frac{R}4}(x_0)}\nu_i(x) h(x)\,d\mathcal{H}^{d-1}(x), \end{aligned}$$

which implies that

$$\displaystyle \begin{aligned} \|\partial_i h\|{}_{L^\infty(B_{\frac{3R}4})}\le \frac{4d}{R}\|h\|{}_{L^\infty(B_{R})}, \end{aligned}$$

and so, we obtain (7.7). Now, by the same argument, we get that

$$\displaystyle \begin{aligned} \|\partial_{ij} h\|{}_{L^\infty(B_{\frac{R}2})}\le \frac{16d^2}{R^2}\|h\|{}_{L^\infty(B_{R})}. \end{aligned}$$

Let now x ∈ B _R∕4 and set

$$\displaystyle \begin{aligned} f(t)=h(xt)\qquad \text{for every}\qquad t\in[0,1]. \end{aligned}$$

Then, we have

$$\displaystyle \begin{aligned} h(x)-h(0)-x\cdot\nabla h(0)=f(1)-f(0)-f'(0)=\int_0^1(1-t)f''(t)\,dt. \end{aligned}$$

Since

$$\displaystyle \begin{aligned} f'(t)=x\cdot \nabla h(xt) \quad \; \text{ and } \quad \; f''(t) = \sum_{i,j=1}^d x_{i}x_{j}\partial_{ij} h(xt), \end{aligned}$$

and

$$\displaystyle \begin{aligned} x\cdot \nabla h(xt)=\sum_{i,j=1}^d\int_0^tx_ix_j\partial_{ij}h(sx)\,ds, \end{aligned}$$

we get precisely (7.8). □

Proof of Theorem 7.4

We fix C ₀ and r ₀ to be dimensional constant which will be chosen later. In order to prove that ε ₀ exists we reason by contradiction. Let ε _n → 0 and let $u_n:B_1\to \mathbb {R}$ be a sequence of continuous functions satisfying the conditions (a), (b) and (c) with ε _n. Without loss of generality, we may suppose that, for any $n\in \mathbb {N}$, u _n is ε _n flat in B ₁ in the same direction e _d. Finally, we assume by contradiction that, there are no $n\in \mathbb {N}$ and a unit vector ν satisfying the following conditions:

(i) |ν − e _d|≤ C ₀ ε;

(ii) the function $(u_n)_{r_0}:B_1\to \mathbb {R}$ is σε-flat in B ₁, in the direction ν.

By Lemma 7.14 we can suppose that the sequence

$$\displaystyle \begin{aligned} \tilde u_n(x)=\frac{u_n(x)-x_d}{{\varepsilon}_n}\quad \text{for}\quad x\in B_{1}\cap \overline \Omega_{u_n}, \end{aligned}$$

converges (in the sense of Lemma 7.14 (i) and (ii)) in B _1∕2 to a smooth (C ^∞(B _1∕2)) function

$$\displaystyle \begin{aligned} \tilde u:B_{{1}/{2}}\cap \{x_d\ge 0\}\to [-1,1] \end{aligned}$$

that satisfies (7.6). We notice that

$$\displaystyle \begin{aligned} \tilde u(0)=0\quad \text{and}\quad \frac{\partial \tilde u}{\partial x_d}(0)=0. \end{aligned}$$

We set

$$\displaystyle \begin{aligned} \displaystyle\nu_i:=\frac{\partial \tilde u}{\partial x_i}(0),\quad \text{for every}\quad i=1,\dots,d-1\ ;\qquad \nu':=(\nu_1,\dots,\nu_{d-1})\in\mathbb{R}^{d-1}, \end{aligned}$$

and we re-write (7.8) as

$$\displaystyle \begin{aligned} \nu'\cdot x'-4C_d |x|{}^2\le \tilde u(x)\le \nu'\cdot x'+4C_d |x|{}^2\qquad \text{for every}\qquad x=(x',x_d)\in B_{{1}/{4}}\cap \{x_d\ge 0\}. \end{aligned}$$

We now fix r ≤ 1∕4. Since the graph Γ_n of $\tilde u_n$ converges in the Hausdorff distance to the graph Γ of $\tilde u$ (see Lemma 7.14 (ii)), we have that for n large enough

$$\displaystyle \begin{aligned} \nu'\cdot x'-8 C_d r^2\le \tilde u_n(x)\le \nu'\cdot x'+8 C_d r^2\qquad \text{for every}\qquad x=(x',x_d)\in B_r\cap \overline \Omega_{u_n}. \end{aligned} $$

(7.9)

Using the definition of $\tilde u_n$ we can rewrite (7.9) as

$$\displaystyle \begin{aligned} x_d+{\varepsilon}_n\nu'\cdot x'-{\varepsilon}_n8 C_d r^2\le u_n(x)\le x_d+{\varepsilon}_n\nu'\cdot x'+{\varepsilon}_n 8 C_dr^2, \end{aligned} $$

(7.10)

which holds for every $x=(x',x_d)\in B_r\cap \overline \Omega _{u_n}$.

We define the new flatness direction ν as follows:

$$\displaystyle \begin{aligned} \displaystyle\nu:=\frac{1}{\sqrt{1+{\varepsilon}_n^2|\nu'|{}^2}}({\varepsilon}_n\nu',1)\,\in\mathbb{R}^d. \end{aligned}$$

By definition, we have that |ν| = 1. We next estimate the distance between ν and e _d. Since both ν and e _d are unit vectors, we have

$$\displaystyle \begin{aligned} |\nu-e_d|{}^2=2(1-\nu\cdot e_d)=2\left(1-\frac 1{\sqrt{1+{\varepsilon}_n^2|\nu'|{}^2}}\right). \end{aligned}$$

Notice that the following elementary inequality holds:

$$\displaystyle \begin{aligned} 1-\frac{1}{\sqrt{1+X}}\le 2X\qquad \text{for every}\qquad -{1}/{2}< X< {1}/{2}\,. \end{aligned} $$

(7.11)

In order to apply this inequality to $X={\varepsilon }_n^2|\nu '|{ }^2$, we first check that ${\varepsilon }_n^2|\nu '|{ }^2\le {1}/{2}$. In fact, by the definition of ν′ and (7.7), we have the estimate |ν′|≤ 2C _d. Thus, for n large enough, we have that ${\varepsilon }_n^2|\nu '|{ }^2\le {1}/{2}$ and so, we can estimate

$$\displaystyle \begin{aligned} {} |\nu-e_d|{}^2\le 2|\nu'|{}^2{\varepsilon}_n^2\le 8C_d^2{\varepsilon}_n^2\,, \end{aligned}$$

which proves that ν satisfies (i), once we choose C ₀ = 4C _d.

Using again the inequality (7.11) and the fact that

$$\displaystyle \begin{aligned} 0\le u_n\le {\varepsilon}_n+r\qquad \text{in}\qquad B_r, \end{aligned}$$

which follows by the non-negativity and the ε _n-flatness of u _n, we get that

$$\displaystyle \begin{aligned} u_n-8C_d^2{\varepsilon}_n^2(r+{\varepsilon}_n)\le \frac{u_n}{\sqrt{1+{\varepsilon}_n^2|\nu'|{}^2}}\le u_n\qquad \text{in}\qquad B_r. \end{aligned}$$

Thus, dividing (7.10) by $\sqrt {1+{\varepsilon }_n^2|\nu '|{ }^2}$, we get that

$$\displaystyle \begin{aligned} {} x\cdot\nu-C_d\big({\varepsilon}_n^2(r+{\varepsilon}_n)+{\varepsilon}_n r^2\big)\le u_n(x)\le x\cdot\nu+C_d {\varepsilon}_n r^2, \end{aligned}$$

for every $x=(x',x_d)\in B_r\cap \overline \Omega _{u_n}$, C _d being a dimensional constant. Choosing r ₀ small enough and ε ₀ ≤ r ₀, we get that

$$\displaystyle \begin{aligned} {} x\cdot\nu-{{\varepsilon}_n r_0}\sigma\le u_n(x)\le x\cdot\nu+{{\varepsilon}_n r_0}\sigma\qquad \text{for every}\qquad x=(x',x_d)\in B_{r_0}\cap \overline \Omega_{u_n}, \end{aligned}$$

and so the vector ν satisfies (i) and (ii), in contradiction with the initial assumption. □