$C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem

TL;DR

This paper proves that the free boundary in a one-phase fractional Laplacian problem is infinitely smooth, extending previous regularity results from $C^{1, ext{alpha}}$ to $C^{ ext{infinity}}$ for all $0<s<1$.

Contribution

The authors establish $C^{ ext{infinity}}$ regularity of the free boundary for the fractional Laplacian problem, generalizing prior $C^{1, ext{alpha}}$ results and extending known results for the case $s=1/2$.

Findings

Free boundary is $C^{ ext{infinity}}$ smooth.

Extension of regularity results from $s=1/2$ to all $0<s<1$.

Improved understanding of free boundary regularity in fractional problems.

Abstract

We consider a one-phase free boundary problem involving fractional Laplacian $(-\Delta)^s$, $0<s<1$. D. De Silva, O. Savin, and Y. Sire proved that the flat boundaries are $C^{1,\alpha}$. We raise the regularity to $C^{\infty}$, extending the result known for $(-\Delta)^{1/2}$ by D. De Silva and O. Savin.

Equations

$$\{\begin{array}{rlrl}& u\ge 0,& & \text{ in }\Omega ,\\ & \Delta u=0,& & \text{ in }\Omega \cap \{u>0\},\\ & |Du|=1,& & \text{ on }\Omega \cap \partial \{u>0\}.\end{array}$$

$$(-\Delta {)}^{s}u(x)=c_{n,s}P.V.{\int }_{{\mathbb{R}}^n}\frac{u(x)-u(x^{\prime })}{|x-x^{\prime }{|}^{n+2s}}d{x}^{\prime },0<s<1,$$

$$\{\begin{array}{rlrl}& u_0\ge 0,& \text{ in }& \Omega ,\\ & (-\Delta {)}^s{u}_0=0,s\in (0,1),& \text{ in }& \Omega \cap \{u>0\},\\ & \underset{t\to 0^{+}}{\lim }\frac{u_0(x+t\nu (x))}{t^s}=1,& \text{ on }& F(u):=\Omega \cap \partial \{u>0\},\end{array}$$

$$\min\left\{\int_{0}^{\infty}\int_{\mathbb{R}^{n}}y^{\beta}|\nabla u(x,y)|^{2}dxdy:\;\beta=1-2s,u\Big{|}_{\mathbb{R}^{n}\times\{0\}}=u_{0}\right\}.$$

$$\{\begin{array}{rl}& L_{\beta }u=0,\text{in }{\mathbb{R}}^n\times (0,+\infty ),\\ & u=u_0,\text{on }{\mathbb{R}}^n\times \{0\},\\ & (-\Delta {)}^s{u}_0(x)=d_{n,s}\underset{y\to 0^{+}}{\lim }\frac{u(x,y)-u(x,0)}{y^{1-\beta }},\end{array}$$

$$L_{\beta }(u):=\text{div}(|y{|}^{\beta }\nabla u).$$

$$\{\begin{array}{rlrl}& u\ge 0,& \text{in}& B_1,\\ & L_{\beta }u=0,& \text{in}& B_1\cap \{u>0\},\\ & \frac{\partial u}{\partial {\nu }^s}=1,& \text{on}& F(u),\end{array}$$

$$\frac{\partial u}{\partial {\nu }^s}(x_0):=\underset{t\to 0^{+}}{\lim }\frac{u(x_0+t\nu (x_0))}{t^s},x_0\in F(u).$$

$$\{x\in B_1:x_n\le -\bar{\epsilon }\}\subset \{x\in B_1:u(x,0)=0\}\subset \{x\in B_1:x_n\le \bar{\epsilon }\},$$

$$\{x\in B_1:x_n\le -\bar{\epsilon }\}\subset \{x\in B_1:u_0(x,0)=0\}\subset \{x\in B_1:x_n\le \bar{\epsilon }\},$$

$$\{x\in B_1:x_n\le -\bar{\epsilon }\}\subset \{x\in B_1:u(x,0)=0\}\subset \{x\in B_1:x_n\le \bar{\epsilon }\},$$

$$\mathcal{P}=\{(x,y):x_n\le 0,y=0\},\mathcal{L}=\{(x,y):x_n=y=0\}.$$

$$F(u)={\partial }_{{\mathbb{R}}^n}({\mathbb{R}}^n\times \{0\}\cap \{u>0\})\cap B_1\subset {\mathbb{R}}^n\times \{0\},$$

$$F(u)=\{(x^{\prime },g(x^{\prime }),0):g(0)=0,g\in C^{1,\alpha },\nabla g(0)=0\},$$

$$x^{\mu }=x_1^{{\mu }_1}\cdots x_n^{{\mu }_n},|\mu |={\mu }_1+\cdots +{\mu }_n,{\mu }_i\ge 0,\Vert P\Vert :=\max |a_{\mu m}|.$$

$$P(x,r)=a_{\mu m}{x}^{\mu }{r}^m,\mathrm{deg}P=k,$$

$$U(t,y)=(\frac{-t+\sqrt{t^2+y^2}}{2}{)}^s.$$

$$U_t=s\frac{U}{r},U_{tt}=\frac{sr-t}{r^2}{U}_t=\frac{s(sr-t)}{r^3}U,r=\sqrt{t^2+y^2}.$$

$$u(X)\ge v(X)(\text{resp.}u(X)\le v(X))\text{in a neighborhood }O\text{ of }{X}_0.$$

$$v(x,y)=aU((x-x_0)\cdot \nu (x_0),y)+o(|(x-x_0,y){|}^s),\text{as}(x,y)\to (x_0,0),$$

$$L_{\beta }u=0\text{in}{B}_1\setminus \{y=0\}.$$

$$f(X)=P_0(x,r)+O(|X{|}^{k+\alpha }).$$

$$\Vert P_0\Vert \le M\text{and}|f(X)-P_0(x,r)|\le M|X{|}^{k+\alpha },$$

$$\Vert f{\Vert }_{C_{xr}^{k,\alpha }(Z)}\le M\text{for all }Z\in K.$$

$$\{\begin{array}{rlrl}& L_{\beta }(u_{n}w)=0,& \text{in }& B_1\cap \{u>0\},\\ & |{\nabla }_{r}w|=0,& \text{on }& B_1\cap F(u).\end{array}$$

$$|{\nabla }_{r}w|(X_0):=\underset{\begin{array}{c}(t,y)\to (0,0)\\ r^2=t^2+y^2\end{array}}{\lim }\frac{w(X_0+t\nu +y{e}_{n+1})-w(X_0)}{r},$$

$$\{\begin{array}{rlrl}& L_{\beta }(U_{n}w)=0,& \text{in }& B_1\setminus \mathcal{P},\\ & |{\nabla }_{r}w|=0,& \text{on }& B_1\cap \mathcal{L},\end{array}$$

$$\varphi (X)=\varphi (X_0)+a(X_0)\cdot (x-c)+b(X_0)r+O(|x-c{|}^2+r^{1+\alpha }),$$

$$|h-(Q(x^{\prime })+rP(x,r))|\le C_o|X{|}^{k+1}\text{ in }{B}_{1/2},$$

$$\Big{|}h(X)-\Big{(}h(0)+\xi_{0}\cdot x^{\prime}+\frac{1}{2}{x^{\prime}}^{T}M_{0}x^{\prime}-\frac{1}{2s}\Big{(}\frac{a_{0}}{2}r^{2}+b_{0}rx_{n}\Big{)}\Big{)}\Big{|}\leq C_{o}|X|^{3}\text{ in }B_{1}\setminus\mathcal{P},$$

Full text

lyuruncao@u.nus.edu

1. 1 Introduction

2. 2 Preliminary

3. 2.1 Notation

4. 2.2 Viscosity solution

5. 2.3 The Linearized Problem

6. 3 A Family of Functions

7. 4 $C^{2,\alpha}$ Regularity

8. 5 $C^{\infty}$ Regularity

9. 6 Expansion Theorem for Linearized Problem

10. 7 Appendix

1. Introduction

Since the 1970s, free boundary problems have become one of the most important subjects in the field of elliptic PDE. One of the most famous problems is the classical one-phase Alt-Caffarelli or Bernoulli problem analyzed by H. Alt and L. Caffarelli in [AC81]. It is

[TABLE]

It is a natural model for long-range interactions appearing in large systems of interacting particles, flame propagation, and probability theory. Those models raised the interest of mathematicians to analyze the regularity of the free boundary of $u$ [CRS10, FRRO24c, DL76]. Specifically, mathematicians aim to find conditions under which the free boundary is smooth. A series of papers [AC81, DSFS19, LZ23, KN77] proved that locally flat free boundaries are $C^{\infty}$. On the other hand, [AC81, Wei98] proved that the set of singular points on the free boundary is small in the sense of Hausdorff dimension. We refer to [CS05, FRRO22, Vel23] for further details in this field. We also refer to some important papers on the regularity of the solutions [ROS16, ROS17, FROS20, STV21a, STV21b, FRO22, FRRO24a, FRRO24b] for some Schauder-type regularity results. Note that the regularity of the free boundary and that of the solution are closely related.

In this paper, we focus on the one-phase fractional Laplacian problem. The fractional Laplacian operator is a standard model for non-local elliptic operators. Over the past two decades, non-local free boundary problems, especially those involving the fractional Laplacian operator, have attracted significant attention [CSS07, CRS10, CF13, CROS17, BFRO18a, BFRO18b, ARO20, ROTLW25, ROW25b].

The fractional Laplacian operator is a Fourier multiplier in $\mathbb{R}^{n}$ with symbol is $|\xi|^{2s}$. Alternatively, we can define the operator by an integral representation:

[TABLE]

where $c_{n,s}$ is a constant depending on $n$ and $s$, and $P.V.$ denotes the Cauchy principal value. In this paper, we consider the one-phase Alt-Caffarelli problem for the fractional Laplacian $(-\Delta)^{s}$, $0<s<1$, in an open domain $\Omega\subset\mathbb{R}^{n}$, following [CRS10, DSSS14, EKP*+*21],

[TABLE]

where $u_{0}$ is defined on the entire space $\mathbb{R}^{n}$ with prescribed values outside $\Omega$. In particular, the case $s=\frac{1}{2}$ was studied in greater detail in [DR12, DS12, DSS12, DS15].

To study the problem, we will use the Caffarelli-Silvestre extension. It plays a key role in our work. As noted earlier, the non-local nature of the fractional Laplacian precludes the direct application of classical local PDE techniques to the function $u_{0}$. However, the work of Caffarelli and Silvestre in [CS07, CS16] allows us to extend $u_{0}$ from $\mathbb{R}^{n}$ to $\mathbb{R}^{n+1}$. Consider the following minimizing problem

[TABLE]

The minimizer $u$ satisfies the following equation:

[TABLE]

where $d_{n,s}$ is a constant depending on $n$ and $s$, and $L_{\beta}$ is defined by

[TABLE]

The operator $L_{\beta}$ is local, allowing us to use standard elliptic techniques. This extension transforms the non-local problem for $u_{0}$ into a local one for $u$. Consequently, it suffices to study the following boundary value problem:

[TABLE]

where $-1<\beta=1-2s<1$, $\nu$ is the interior normal unit vector of $F(u)$, and

[TABLE]

For general $0<s<1$, the optimal regularity of solutions has been established in [CRS10, ROW24]. The $C^{1,\alpha}$ regularity for flat boundaries was established in [DR12, DSSS14, ROW25a]. The smallness of the singular set was proven in [EKP*+*21]. Specifically, [DS15] improved the regularity of flat free boundary for $s=\frac{1}{2}$ to $C^{\infty}$.

As noted in [ROW25a, Remark 1.2], the best-known regularity result for the free boundary when $s\neq\frac{1}{2}$ is $C^{1,\alpha}$. A recent paper [BROW25] proved that $C^{2,\alpha}$ free boundaries are $C^{\infty}$ for general non-local operators. We state the $C^{1,\alpha}$ regularity result by D. De Silva, O. Savin, and Y. Sire. Our work is based on that regularity result.

Theorem 1.1**.**

([DSSS14, Theorem 1.1]) There exists a small constant $\bar{\varepsilon}>0$ depending on $n$ and $\beta$, such that if $u$ is a viscosity solution to $(\ref{extension of fracional laplacian})$ satisfying

[TABLE]

then $F(u)$ is $C^{1,\alpha}$ in $B_{1/2}$. Here $\alpha$ depends on $n,\beta$.

In this paper, we improve the regularity of flat free boundary from $C^{1,\alpha}$ to $C^{\infty}$. It is our main Theorem.

Theorem 1.2**.**

There exists a small constant $\bar{\varepsilon}>0$ depending on $n$ and $s$, such that if $u_{0}$ is a viscosity solution to $(\ref{fractional laplacian})$ satisfying

[TABLE]

then the free boundary $F(u_{0})$ is $C^{\infty}$ in $B_{1/2}$.

Using Caffarelli-Silvestre extension, the fractional Laplacian can be extended to $\mathbb{R}^{n+1}$. Throughout this paper, we mainly consider the extended problem. We will prove the extended version of Theorem 1.2:

Theorem 1.3**.**

There exists a small constant $\bar{\varepsilon}>0$ depending on $n$ and $\beta$, such that if $u$ is a viscosity solution to $(\ref{extension of fracional laplacian})$ satisfying

[TABLE]

then $F(u)$ is $C^{\infty}$ in $B_{1/2}$.

Remark 1.4**.**

Our strategy for proving the smoothness of $C^{1,\alpha}$ free boundaries consists of two steps. In step 1, we prove that $C^{1,\alpha}$ free boundaries are $C^{2,\alpha}$. In step 2, we prove that $C^{2,\alpha}$ free boundaries are $C^{\infty}$. Step 2 is also recently proved in the paper [BROW25] by B. Barrios, X. Ros-Oton, and Marvin Weidner by considering a general type of global operators. In this paper, we prove by using the Caffarelli-Silvestre extension. This approach is significantly different from the proof in [BROW25]. Our approach is inspired by the work of De Silva and Savin [DS12, DS15]. The techniques in [JN17, STV21a, STV21b, FRO22, FRRO24c] by M. Fall, X. Fernández-Real, Y. Jhaveri, R. Neumayer, X. Ros-Oton, Y. Sire, S. Terracini, and S. Vita provided key insights into the regularity theory.

Now we introduce our strategy in detail. In step 1, we prove following following the spirit of [DS12]. We use a sequence of paraboloids to approximate the free boundary. For each paraboloid, we use function $V_{\mathcal{S},a,b}$ to approximate the solution $u$. During the approximation, we need to analyze the difference of the approximating function $V_{\mathcal{S},a,b}$ and the solution $u$. For $\beta=0$, the limit is still a harmonic function with natural regularity estimate. For general $\beta$, we use Schauder-type estimates to derive analogous results.

In step 2, we prove following following the spirit of [DS15]. The function $\frac{u_{i}}{u_{n}}$ is an extension of the derivatives of $g_{i}$ in (2.2). We prove that it has the same regularity as $g$. The desired result will follow. For general $\beta\in(-1,1)$, we first analyze the linearized problem (2.6) and find polynomial approximation for the solution. That needs us to analyze the degenerate operator $L_{\beta}(U_{n}\cdot)$. The result together with the compactness allows to appoximate the function $\frac{u_{i}}{u_{n}}$.

We notice that step 1: $C^{1,\alpha}\to C^{2,\alpha}$ is essential. The technique in step 2 cannot be applied to $C^{1,\alpha}$ boundaries. To use the technique, we need to analyze the linearized problem with a Neumann boundary condition. However, as [DS15] indicates, the Neumann condition cannot be verified for $C^{1,\alpha}$ boundary.

We follow the spirit of [DS12, DS15] to construct our proof. Compared to the proof for $\beta=0$, there are several difficulties. For $\beta=0$, the operator $L_{\beta}=\Delta$. It has been fully analyzed. For general $\beta\in(-1,1)$, we use the regularity theory about $L_{\beta}$. Roughly speaking, we will consider the regularity theory for the operator $\Delta+\frac{\beta}{y}\partial_{y}$ since $L_{\beta}=|y|^{\beta}(\Delta+\frac{\beta}{y}\partial_{y})$. Secondly, harmonic functions remain harmonic under holomorphic map. Therefore to analyze a solution $w$ to $\Delta(U_{n}w)=0$ in $\mathbb{R}^{n}\setminus\mathcal{P}$, we can consider the slice of the space $\{x^{\prime}=0\}$ and map the 2D slit domain to the upper half plane with a holomorphic map. For general $\beta$, we will use boundary Harnack inequality in slit domain to handle it.

The paper is organized as following. In Section 2, we recall some preliminary results and introduce some notations. In Sections 3 to 4, we raise the regularity to $C^{2,\alpha}$. In Section 3, we introduce a family of functions such that their free boundary are paraboloids. In Section 4, we use the functions to approximate the solution $u$ to prove the $C^{2,\alpha}$ regularity. In Section 5, we prove $C^{\infty}$ regularity by analyzing the linearized problem to show that the free boundary $F(u)$ has the same regularity as its derivatives. In Section 6, we consider the linearized problem with $F(u)$ being a codimension 2 hyperplane. In Section 7, we provide some known results about this problem.

The regularity of flat free boundaries plays a key role in understanding Bernstein-type problems. We are preparing a follow-up paper based on [EFRY23] to analyze graphical solutions. This will help us find an equivalent formulation for Bernstein-type problems.

Acknowledgments: The author would like to thank Hui Yu and Ovidiu Savin for fruitful discussions regarding this project. We refer to a recent paper [BROW25] by B. Barrios, X. Ros-Oton, and Marvin Weidner. It proved that $C^{2,\alpha}$ free boundaries are smooth differently. They also developed the smoothness for obstacle problem.

2. Preliminary

In this section, we recall some definitions and gather some results about the fractional Laplacian problem. First, we introduce the notation used in this paper.

2.1. Notation

In this paper, we consider the problem (1.5) in $\mathbb{R}^{n+1}$. For $X\in\mathbb{R}^{n+1}$, we use $x^{\prime}$ and $x$ to denote its first $n-1$ and $n$ coordinates. We use $x_{n}$,$y$ to denote the $n$-th, $(n+1)$-th coordinates and write $X=(x,y)=(x^{\prime},x_{n},y)$.

In $\mathbb{R}^{n+1}$, a ball with radius $r$ and center $X$ is denoted by $B_{r}(X)$. For simplicity we use $B_{r}$ to denote $B_{r}(0)$. We use $B_{r}^{n}$ to denote the $n$-dimensional ball $B_{r}\cap\{y=0\}$.

The half-plane $\mathcal{P}$ and codimension-two space $\mathcal{L}$ will frequently appear. We define them

[TABLE]

For a continuous non-negative function $u:\mathbb{R}^{n+1}\to\mathbb{R}_{+}$, we define its free boundary $F(u)$ as

[TABLE]

Specifically, we focus on $F(u)$ satisfying:

[TABLE]

where $e_{n}$ is the unit vector in the $n$-th direction.

Throughout this paper, $n,\beta$ are called universal constants. We say a variable is universal if it only depends on universal constants. We say a variable has universal bound if it is bounded by a universal constant.

In this paper, we consider a different version of regularity following [DS15]. Using the notation

[TABLE]

We denote by

[TABLE]

a polynomial of degree $k$ in the $(x,r)$ variables.

2.2. Viscosity solution

Before considering viscosity solutions, we introduce the function $U:\mathbb{R}^{2}\to\mathbb{R}$. Set

[TABLE]

For $t=x_{n}$, $U\in C^{2}(\mathbb{R}^{n+1}\setminus\mathcal{P})$, and $U(x_{n},y)$ is a classical solution to (1.5). Furthermore, the free boundary $F(U)=\mathcal{L}$ and $\{U=0\}=\mathcal{P}$. Therefore $U$ is denoted by half-plane solution. We use $U_{t}$ to denote $\frac{\partial U}{\partial t}$. The following equalities will be of use in this paper.

[TABLE]

Now we construct the definition of viscosity solutions. It allows us to consider a solution to be less regular. To begin with, we state the definitions of touching functions following [DSSS14].

Definition 2.1**.**

For continuous functions $u$ and $v$, we say $v$ touches $u$ from below (resp. above) at $X_{0}\in B_{1}$ if $u(X_{0})=v(X_{0})$ and

[TABLE]

If this inequality is strict in $O\setminus\{X_{0}\}$, we say that $v$ touches $u$ strictly by below (resp. above).

With the concept of touching functions, we give the definition of comparison sub/sup-solutions.

Definition 2.2**.**

A function $v\in C(B_{1})$ is a comparison subsolution to (1.5) if $v$ is non-negative in $B_{1}$, even with respect to $\{y=0\}$, $C^{2}$ in the set $\{v>0\}$, and satisfies

(i)  $L_{\beta}v\geq 0\quad\text{in}\quad B_{1}\setminus\{y=0\},$

(ii) $F(v)$ is $C^{2}$ and if $x_{0}\in F(v)$ we have

[TABLE]

with $a\geq 1$, where $\nu(x_{0})$ denotes the unit normal at $x_{0}$ to $F(v)$ pointing toward $\{v>0\}$;

Moreover, $v$ is a strict comparison subsolution if either (i) holds strictly or $a>1$. Similarly, one can define a (strict) comparison supersolution.

Finally, we are able to give the definition of viscosity solutions to end this subsection following [CC95, DSSS14]. The idea of the definition is that solutions cannot be touched from below/above by sub/sup-solutions. It needs less regularity of $F(u)$ and uniform convergence implies that the limit is close to a solution. Throughout this paper, all solutions are viscosity solutions unless otherwise specified.

Definition 2.3**.**

A function $u$ is a viscosity solution to (1.5) if $u$ is continuous, non-negative in $B_{1}$, and satisfies:

(i) $u$ is locally $C^{1,1}$ in $B_{1}\cap\{u>0\}$, even with respect to $\{y=0\}$ and solves (in the viscosity sense)

[TABLE]

(ii) Any (strict) comparison subsolution (resp. supersolution) cannot touch $g$ from below (resp. by above) at a point $X_{0}=(x_{0},0)\in F(u)$.

For this equation, we need to consider a different type of regularity. Now we give the definition of $f\in C^{k,\alpha}_{xr}$. For usually regularity, we use polynomials of $(x.y)$ to approximate the function. Using $P(x,r)$ to approximate the function induces $C^{k,\alpha}_{xr}$ regularity.

Definition 2.4**.**

For $F(u)$ satisfying (2.2), we say that a function $f:B_{1}\subset\mathbb{R}^{n+1}\to\mathbb{R}$ is pointwise $C^{k,\alpha}$ in the $(x,r)$-variables at $0\in F(u)$, and write $f\in C^{k,\alpha}_{xr}(0)$, if there exists a (tangent) polynomial $P_{0}(x,r)$ of degree $k$ such that

[TABLE]

We define $\|f\|_{C^{k,\alpha}_{xr}(0)}$ as the smallest constant $M$ such that

[TABLE]

for all $X$ in the domain of definition. Furthermore, we say that $f\in C^{k,\alpha}_{xr}(K)$ for $K\subset F(u)$ if there exists a constant $M$ such that $f\in C^{k,\alpha}_{xr}(Z)$ for all $Z\in K$, and

[TABLE]

The smallest such constant $M$ is denoted by $\|f\|_{C^{k,\alpha}_{xr}(K)}$.

2.3. The Linearized Problem

In this subsection, we introduce here the linearized problem associated to (1.5). To improve the regularity of the free boundary $F(u)$, we will show that $\frac{u_{i}}{u_{n}}$, which extends the derivatives of $F(u)$, has the same regularity as $F(u)$. They are solutions to the linearized problem.

For $u$ being a solution to (1.5) satisfying (2.2), the linearized problem associated to (1.5) is

[TABLE]

Here $|\nabla_{r}w|$ denotes the radial derivative, defined as

[TABLE]

for $\nu$ being the normal unit of $F(u)$ at $X_{0}$.

A special case of (2.5) is when $u$ is half-plane solution and $F(u)=\mathcal{L}$. In this case, $u$ is substituted by the half-plane solution $U$. To deal with that, we need to consider the viscoisity type solution. We introduce the following problems following the spirit of [DSSS14].

Definition 2.5**.**

We say $w$ is a viscosity solution to

[TABLE]

if $w\in C^{1,1}_{loc}(B_{1}\setminus\mathcal{P})$, $w$ is even with respect to $\{y=0\}$, and $w$ satisfies:

(i) $L_{\beta}(U_{n}w)=0$ in $B_{1}\setminus\{y=0\}$;

(ii) Let $\phi$ be a continuous function around $X_{0}=(c,0,0)\in B_{1}\cap\mathcal{L}$ satisfying

[TABLE]

for some $\alpha>0$. If $b(X_{0})>0$, then $\phi$ cannot touch $w$ from below at $X_{0}$, and if $b(X_{0})<0$ then $\phi$ cannot touch $w$ from above at $X_{0}$.

The solution to (2.6) has a polynomial expansion at points in $B_{1}\cap\mathcal{L}$. We have the following theorem allowing use to use polynomials to approximate the solution following the spirit of [DSSS14, Theorem 6.1].

Theorem 2.6**.**

Given a boundary data $\bar{h}\in C(\partial B_{1})$ even with respect to $\{y=0\}$, $|\bar{h}|\leq 1$, there exists a unique classical solution $h$ to (2.6) such that $h\in C(\bar{B}_{1})$ is even with respect to $\{y=0\}$ and satisfies $h=\bar{h}$ on $\partial B_{1}$. For any $k\in\mathbb{N}$, there exist $Q$ being order $k$ polynomial in $x^{\prime}$ and $P$ being order $k-1$ polynomial in $x,r$ such that

[TABLE]

for $C_{o}$ depending on universal constants and $k$. Furthermore, $\|Q\|,\|P\|$ have universal bounds and $L_{\beta}(U_{n}(Q+rP))=0$ in $B_{1}\setminus\mathcal{P}$.

In particular, for $n=2$ we have

[TABLE]

for some $a_{0},,b_{0}\in\mathbb{R},\xi_{0}\in\mathbb{R}^{n-1},M_{0}\in S^{(n-1)\times(n-1)}$ with

[TABLE]

and

[TABLE]

3. A Family of Functions

In this section, we introduce a family of functions $V_{\mathcal{S},a,b}$ following the spirit of [DS12] by D. De Silva and O. Savin. They serve as approximate solutions to (1.5) with free boundary $\mathcal{S}$. We mainly study the case $\mathcal{S}$ being a paraboloid.

Given a surface $\mathcal{S}=\{x_{n}=g(x^{\prime})\}\subset\mathbb{R}^{n}$. For any $X=(x,y)\in\mathbb{R}^{n+1}$, we set

[TABLE]

Next we introduce the concept of domain variation functions. It will be used to measure the distance between $v$ and $U$, especially for $v=V_{\mathcal{S}}$ close to $U$. For a function $v$ close to the half-plane solution $U$, we construct the the domain variation function $\tilde{v}$. In Section 4 we will construct a sequence of functions approximating the solution $v$. Domain variations will be an important role in the approximating process.

Definition 3.1**.**

Let $\epsilon>0$ and let $v$ be a continuous non-negative function in $\overline{B_{\rho}}$. We follow [DS12] to define the multivalued map $\tilde{v}_{\epsilon}:\mathbb{R}^{n+1}\setminus\mathcal{P}\to\mathbb{R}$ via the formula

[TABLE]

We call $\tilde{v}_{\epsilon}$ the $\epsilon$-domain variation associated to $v$. By abuse of notation, from now on we write $\tilde{v}_{\epsilon}(X)$ to denote any of the values in this set.

If

[TABLE]

then we have $\tilde{v}_{\epsilon}(X)\neq\emptyset$ for $X\in B_{\rho-\epsilon}\setminus\mathcal{P}$ and $|\tilde{v}_{\epsilon}|\leq 1$. Moreover, if $v$ is strictly monotone in the $e_{n}$-direction in $B_{\rho}(v)\cap\{v>0\}$, then $\tilde{v}_{\epsilon}$ is single-valued.

Domain variations also have the comparison principle. It indicates that domain variation away from the free boundary will control its behavior near the free boundary in some way.

Lemma 3.2**.**

([DSSS14, Lemma 3.1]) Let $u,v$ be respectively a solution and a subsolution to (1.5) in $B_{2}$, with $v$ strictly increasing in the $e_{n}$-direction in $B_{2}(v)\cap\{v>0\}$. Assume that $u$ satisfies the flatness assumption (3.3) in $B_{2}$ for $\epsilon>0$ small and that $\tilde{v}_{\epsilon}$ is defined in $B_{2-\epsilon}\setminus\mathcal{P}$ and bounded. If

[TABLE]

then

[TABLE]

In fact, the behavior of free boundary can imply the property of the domain variation. It can be understood as following: the regularity of free boundary we derive the regularity of solution near it. For a weaker condition of free boundary, we are supposed to find a similar result for the solution. We have:

Lemma 3.3**.**

([DSSS14, Lemma 2.10]) Assume $u$ solves (1.5). Given any $\epsilon>0$ there exists $\delta>0$ and $\bar{\epsilon}>0$, depending on $\epsilon$ such that if

[TABLE]

then the rescaling $u$ satisfies

[TABLE]

We use the notation

[TABLE]

for $\mathcal{S}:=\{x_{n}=\frac{1}{2}(x^{\prime})^{T}Mx^{\prime}+\xi^{\prime}\cdot x^{\prime}\}$, $M$ being a symmetric $(n-1)\times(n-1)$ matrix, and $\xi^{\prime}\in\mathbb{R}^{n-1}$. This is the main case under consideration in this paper. Furthermore, we will investigate $V_{M,\xi^{\prime},a,b}(X)$ with some certain conditions. We give the definition below.

Definition 3.4**.**

For $\delta>0$ small, we define the following classes of functions

[TABLE]

We define the rescaling $V_{\lambda}$ of $V_{M,\xi^{\prime},a,b}$ to be

[TABLE]

To use $\mathcal{V}_{\delta}$ to approximate solution to (1.5), we need to compare them with solutions. In the following proposition, we provide a condition such that $V\in\mathcal{V}_{\delta}$ is a subsolution/supersolution.

Proposition 3.5**.**

Let $V=V_{M,\xi^{\prime},a,b}\in\mathcal{V}_{\delta}$, with $\delta\leq\delta_{0}(n,\beta)$. There exists a constant $C_{0}>0$ depending only on $n,\beta$ such that if

[TABLE]

then $V$ is a comparison subsolution to (1.5) in $B_{2}$.

Proof.

We can easily verify the boundary condition for $V$. Our main task, therefore, is to show that $L_{\beta}V\geq 0$ in $B_{2}$. To this end, we now calculate the action of the operator on $V$. We begin by recalling that

[TABLE]

For the case $\mathcal{S}:=\{x_{n}=\frac{1}{2}(x^{\prime})^{T}Mx^{\prime}+\xi^{\prime}\cdot x^{\prime}\}$, we have

[TABLE]

Applying (3.8) and (3.9), we have

[TABLE]

for $X=(x,y)\in B_{2}$. By analysis we have $|\kappa(x)-trM|\leq C\delta^{2}$ for $C$ depending only on $n,\beta$. For completeness, we show the calculation below:

[TABLE]

Since $r\leq 2$ and $|\xi^{\prime}|,\|M\|\leq\delta$, we have

[TABLE]

Therefore, we obtain the following estimate

[TABLE]

for $C_{0}$ universal. If $a+b-trM\geq C_{0}\delta^{2}$ we have the desired result. ∎

Next we derive the comparison between $V_{n}$,$L_{\beta}V$ and $U_{n}$ for $\delta$ small. It indicates that $V$ is close to $U$ for $\delta$ small. Furthermore, we have a quantitative estimate for $V_{n},L_{\beta}V$. We will use it for the comparison between $V$ and $u$.

Proposition 3.6**.**

Let $V=V_{M,\xi^{\prime},a,b}\in\mathcal{V}_{\delta}$ with $\delta\leq\delta_{0}$, $\delta_{0}$ universal. Here $\mathcal{V}_{\delta}$ is defined in Definition 3.4. We have

[TABLE]

here $\mathcal{P}$ is defined in (2.1). If $V\in\mathcal{V}_{\delta}^{0}$, then

[TABLE]

Proof.

As (3.13) is a direct result of (3.11) and (3.12), it suffices to prove (3.12). In step 1 we establish the proportionality between $V_{n}$ and $U_{t}(t,y)$, where $t$ defined in (3.1). In step 2 we substitute $t$ by $x_{n}$.

Step 1: Proportionality between $V_{n}$ and $U_{t}(t,y)$.

By defintion, we have

[TABLE]

We apply (3.10) to compare $\partial_{t}v_{a,b}$ and $U_{t}$ to obtain

[TABLE]

Step 2: Substitute $t$ by $x_{n}$.

It suffices to derive the comparison between $U_{t}(t,y)$ and $U_{n}(x_{n},y)$. From Harnack inequality, we have

[TABLE]

For the domain given in (3.12), we have

[TABLE]

Therefore $B_{|t-x_{n}|}(x_{n},y)\subset B_{2|t-x_{n}|}(x_{n},y)\subset\{U>0\}$. Applying (3.15) for $t_{1}=t$ and $t_{2}=x_{n}$, the proof is done. ∎

The following proposition gives an estimate of $\tilde{V}$ for $V\in\mathcal{V}_{\delta}$. For $\delta$ small, we know that the domain variation is close to [math]. As the Proposition above, the following proposition gives a quantitative estimate for it. This will be an important tool in our paper.

Proposition 3.7**.**

Let $V=V_{M,\xi^{\prime},a,b}\in V_{\delta}$, with $\delta\leq\delta_{0}$ universal. Then $V$ is strictly monotone increasing in the $e_{n}$-direction in $B_{2}^{+}(V)$. Moreover, $V$ satisfies

[TABLE]

\text{with }r=\sqrt{x_{n}^{2}+y^{2}}\text{ and }C_{1}\text{ depending only on n,\beta}.

Proof.

The monotonicity of $V$ follows from (3.14). The proof of the inequality is similar to the proof of Proposition 3.6. In step 1 we derive the estimate for $t$. It is quite the case in 2D space such that we do not need to consider the terms about $x^{\prime}$. In step 2 we establish the estimate (3.16) by considering the difference between $t+\gamma_{a,b}(t,s)$ and $x_{n}+\gamma_{V}(X)$.

Step 1: Estimate for $t$.

In this step we prove that $v_{a,b}$ satisfies

[TABLE]

for $C$ depending only on $n,\beta$.

To begin with, we apply the Taylor expansion to $U$ and (2.4) to derive

[TABLE]

By (2.4), we can estimate of $U_{t}$ and $U_{tt}$ to obtain:

[TABLE]

for some $t^{\prime}$ satisfying $t\leq t^{\prime}\leq t+\mu$ and $|\mu|\leq 2/r$.

Choosing $\mu=\tilde{\mu}\pm 2\frac{C\tilde{\mu}^{2}}{sr}$, we obtain

[TABLE]

provided that $|\tilde{\mu}/r|<c(n,\beta)$ sufficiently small. Set $\tilde{\mu}=\frac{1}{2s}(\frac{a}{2}r^{2}+btr)$ in (3.18), then (3.17) is obtained.

Step 2: Proof of (3.16).

In this step, we will prove

[TABLE]

We notice

[TABLE]

Therefore it suffices to prove

[TABLE]

then (3.19) follows by applying the inequality to (3.17). We will prove the (3.20) by comparing $t$ and $x_{n}$. We have

[TABLE]

Integrating this inequality on the segment $(x^{\prime},h(x^{\prime})),(x^{\prime},x_{n})$, we obtain

[TABLE]

In $B_{1}$, the surface $\mathcal{S}$ and $x_{n}=0$ are within distance $\delta$ from each other. We have $|t-x_{n}|\leq C\delta$. Hence

[TABLE]

The last two inequalities prove (3.19). The monotonicity of $U$ in the $e_{n}$ direction gives the estimate for $\tilde{V}$. ∎

Remark 3.8**.**

Proposition 3.7 can give an estimate of $V$ by half-plane solution $U$. If $V\in\mathcal{V}_{\delta/4}$, then $V$ satisfies

[TABLE]

We call (3.21) the $\delta-$flatness assumption in $B_{1}$.

Next we vaompare the functions $V$. In the following lemma, we show that the comparison of two surfaces will imply the comparison of two functions.

Lemma 3.9**.**

Let $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ be surfaces with boundaries of curvature not exceeding 2. Let

[TABLE]

Assume that

[TABLE]

with $g_{i}$ Lipschitz graphs, $g_{i}(0)=0$, $|\nabla g_{i}|\leq 1$ and $c$ depending only on $n,\beta$. If

[TABLE]

for some small $\epsilon\leq c$, then for some universally big $C$ we have

[TABLE]

Proof.

After a rescaling of factor $1/\sigma$, it suffices to prove our lemma for $\sigma=1$ and with the curvature of $S_{i}$, $a_{i}$, $b_{i}$ and $\epsilon$ smaller than $c$ universal. This proof is similar to the proofs above. In step 1 we prove the desired result for $t$. It is quite the 2D space case. In step 2 we establish the desired result by considering the difference between dist$(X,\mathcal{S}_{i})$, $i=1,2$.

Step 1: Estimate for $t$.

In this step we prove that for $0<\lambda\leq 1$,

[TABLE]

By (3.14), $\partial_{t}v_{a,b}$ is proportional to $\partial_{t}U$ in the disk of radius 2. Since on the segment with endpoints $(t,y)$ and $(t+C\epsilon\lambda^{2},y)$ all the values of $\partial_{t}U$ are comparable (see (3.15)) we obtain (using $rU_{t}=sU$)

[TABLE]

and our claim is proved.

Step 2: Estimate for dist$(X,\mathcal{S}_{i})$, $i=1,2$

Since $v_{a_{2},b_{2}}$ is increasing in the first coordinate, we obtain that

[TABLE]

On the other hand, from the hypotheses on $h_{i}$ we see that

[TABLE]

where $\bar{t}_{2}$ is the distance to $\mathcal{S}_{2}-C^{\prime}e_{n}$, for some $C^{\prime}$ large depending on the $C$ above. Hence in $B_{1}$ we have

[TABLE]

∎

By the preparation above, we now seek for Harnack-type Theorem 3.12 following the spirit of [DS12]. It will be an important tool in Section 4. To begin with, we prove the following lemma. It shows that the comparison of $u$ and $V$ at a given point indicates the comparison near the free boundary.

Lemma 3.10**.**

If $u$ solves (1.5) and

[TABLE]

[TABLE]

with $V=V_{M,\xi^{\prime},a,b}\in\mathcal{V}_{\delta}^{0}$, $\bar{C}\delta^{2}\leq\epsilon$ for $\bar{C}>0$ universal, then

[TABLE]

[TABLE]

for $\tau\in(0,1)$ universal.

Proof.

We consider Taylor expansion of $V$ at $X-\epsilon e_{n}$. It gives that for any $\tau\in(0,1)$, there exists $\lambda\in(0,1)$ such that

[TABLE]

By (3.12) and (3.15), $V_{n}(X+\lambda\epsilon e_{n})$ is comparable to $U_{n}(X)$. We derive

[TABLE]

Set $h(X):=u(X)-V(X-\epsilon e_{n})$. It suffices to show

[TABLE]

With (3.23) and (3.24), we have the desired result by setting $\tau=c_{1}/C_{1}$.

We prove (3.24) in three steps. In step 1, we derive the equations satisfied by $h$. In step 2, we prove it for values of $h$ in $B_{1/8}(\frac{1}{4}e_{n})$ using Theorem 7.1 Harnack inequality, Theorem 7.1. In step 3, we prove it for values out of $B_{1/8}(\frac{1}{4}e_{n})$ using Theorem 7.2 boundary Harnack inequality, Theorem 7.2.

Step 1: Equations satisfied by $h$.

In this step, we estimate $h(\bar{X})$ and $L_{\beta}h$. Based on (3.22), we have $h\geq 0$ in $B_{1}$ and

[TABLE]

Notice that the same as (3.23), the second inequality is derived by (3.12) and (3.15).

Next we estimate $L_{\beta}h=|y|^{\beta}(\Delta+\frac{\beta}{y}\partial_{y})h$. Based on (3.13), we have

[TABLE]

where $\mathcal{P}$ is the half plane in (2.1) and $\tilde{\mathcal{C}}\supset\supset\mathcal{C}$ is the $\frac{\sqrt{s}d}{2}$-neighborhood of $C$.

Step 2: Values of $h$ in $B_{1/8}(\frac{1}{4}e_{n})$.

We set $h=h_{1}+h_{2}$ satisfying

[TABLE]

and

[TABLE]

Consider $\varphi(X)=-|X-\frac{1}{4}e_{n}|^{2}+(\frac{3}{16})^{2}$. We have $\varphi=0$ on $\partial(B_{3/16}(\frac{1}{4}e_{n}))$ and $L_{\beta}\varphi=-2(n+1+\beta)|y|^{\beta}$. By maximum principle, we have $h_{2}\leq\frac{C\delta^{2}}{2(n+1+\beta)}\varphi\leq C\delta^{2}$. Apply interior Harnack inequality to $h_{1}$, we have that $h_{1}\geq c_{2}\epsilon$ in $B_{1/8}(\frac{1}{4}e_{n})$, for some universal $c_{2}>0$.

Finally, we estimate $h$ to end this step. For $\bar{C}$ large enough, $\delta^{2}$ is much smaller than $\epsilon$. Therefore we have:

[TABLE]

Step 3: Values of $h$ out of $B_{1/8}(\frac{1}{4}e_{n})$.

Now we proceed to estimate $h$ out of $B_{1/8}(\frac{1}{4}e_{n})$. In this domain, the values near $\mathcal{P}$ are more sophicated. We will use Theorem 7.2 boundary Harnack inequality to estimate it. As in step 2, the boundary and $L_{\beta}h$ effect $h$ in opposite ways. We will use two functions $q_{1}$ and $q_{2}$ to estimate them. Set

[TABLE]

where $\mathcal{P}$ as in (2.1) and $\tilde{\mathcal{C}}$ is the $\frac{\sqrt{s}d}{2}$-neighborhood of $\mathcal{C}$. To handle boundary effects, we construct auxiliary functions $q_{1},q_{2}$ satisfying

[TABLE]

with boundary conditions respectively

[TABLE]

[TABLE]

By Theorem 7.2 boundary Harnack inequality, $q_{1}$ is comparable to $q_{2}+\frac{y^{2}}{2(1+\beta)}$ in a neighborhood of $\mathcal{P}\cap\mathcal{C}\subset\subset\tilde{\mathcal{C}}$. For the points away from $\mathcal{P}$, we can simply use interior Harnack to derive $q_{1}\lesssim q_{2}$ as in step 2. Then we have

[TABLE]

with $c_{4}>0$ universal. By the maximum principle,

[TABLE]

since $h\geq q$ on $\partial D$ and $\Delta h\leq\Delta q$ in $D$. Hence, by (3.28) we have

[TABLE]

where in the last inequality we used that $q_{1}$ and $U_{n}$ are comparable. We obtain this as a direct result of Theorem 5.12 boundary Harnack inequality. ∎

With the lemma above, we prove the following proposition. It somehow indicates that if $F(u)$ is between two paraboloids, then we can improve the approximation in a smaller ball. It is similar to the improvement of flatness in [Sil11] by D. De Silva.

Proposition 3.11**.**

There exist $\bar{\epsilon},\bar{\delta}>0$ universally small and $C>0$ universally big, such that if $u$ solves (1.5) and it satisfies

[TABLE]

with

[TABLE]

then either

[TABLE]

or

[TABLE]

for a small universal constant $\eta\in(0,1)$. Here $\mathcal{V}_{\delta}^{0}$ is defined in Definition 3.4.

Proof.

Without loss of generality, we assume

[TABLE]

and prove (3.30). For the case $u(\bar{X})-V(\bar{X})\leq 0$, we can prove (3.29) in the same way. We compare $\tilde{u}$ and $\tilde{V}$ in this proof rather than considering the comparison of $u$ and $V$ directly. To begin with, we apply Lemma 3.10 to derive:

[TABLE]

where $\mathcal{P}$ is defined in (2.1) and $\mathcal{C}^{\prime}:=\{(x^{\prime},x_{n},y):\sqrt{s}d\leq|(x_{n},y)|\leq\frac{1}{4},|x^{\prime}|\leq\frac{1}{2}\},\;d:=\frac{1}{8\sqrt{n-1}}.$

We prove (3.31) in 2 steps. In step 1, we construct a subsolution $\tilde{W}$ close to $\tilde{u}$. In step 2, we transfer the comparison between $\tilde{u}$ and $\tilde{W}$ to the comparison between $\tilde{u}$ and $\tilde{V}$.

Step 1: Subsolution $W$.

In this step, we perturb $V$ to construct $W$, ensuring it remains a subsolution while capturing the error terms. Then we can apply Lemma 3.2. We set

[TABLE]

for $c$ small enough to be chosen. As $\frac{a}{2s}+(2c\epsilon+b)-(\text{tr}(M)+c\epsilon)=c\epsilon\geq C_{0}(\delta+\epsilon)^{2}$, $W$ is a subsolution by Proposition 3.5.

By Proposition 3.7, we estimate the difference of $\tilde{V}$ and $\tilde{W}$

[TABLE]

By choosing $\bar{C}$ big and $c,\bar{\epsilon}$ small such that $4C_{1}\bar{\epsilon}+4C_{1}/\bar{C}+2c/s\leq\frac{\tau}{2}$, we have

[TABLE]

Similarly, by choosing $\bar{C}$ big such that $\tau^{*}:=7cd^{2}-4C_{1}/\bar{C}-4C_{1}\epsilon$ satisfying $0<\tau^{*}<\tau/2$, we have

[TABLE]

Together with (3.31), we have

[TABLE]

We apply Lemma 3.2 and conclude

[TABLE]

Step 2: Compare $\tilde{V}$ and $\tilde{W}$.

In step 1, we have proven (3.33). It suffices to substitute $\tilde{W}$ by $\tilde{V}$. From (3.32), we have that

[TABLE]

$\text{if }c\frac{(2\eta)^{2}}{2(n-1)}+4C_{1}/\bar{C}+4C_{1}\bar{\epsilon}\leq\frac{\tau^{*}}{2}\text{ and }2\eta<\sqrt{s}d.$ The inequality is satisfied by choosing $\bar{C}$ big and $\bar{\epsilon}$ small. Combine the above result with (3.33), we have

[TABLE]

Finally, we prove the desired result. From (3.19) and $V$ being monotone, $u,$ $V$ are $(4\delta+\epsilon)$-flat in $B_{1}$ as in Remark 3.8. For any $X\in B_{\eta}$, we set $\exists Y=X+\tilde{u}(Y)e_{n}$. We require $\eta\geq 4\bar{\delta}+\bar{\epsilon}$ such that $Y\in B_{2\eta}$. The desired result follows since

[TABLE]

∎

With the proposition above, we naturally wish to raise the approximation inductively. We cannot raise the approximation infinitely because that each time we raise it, the surface is farther from plane. The following theorem describes how much we can raise the approximation in a given ball.

Theorem 3.12**.**

There exist $\bar{\epsilon}>0$ small and $\bar{C}>0$ large universal, such that if $u$ solves (1.5) and it satisfies

[TABLE]

with $V=V_{M,\xi^{\prime},a,b}\in\mathcal{V}^{0}_{\delta}$ and

[TABLE]

with $|a_{0}|,|b_{0}|\leq 1$, then

[TABLE]

with

[TABLE]

for a small universal constant $\bar{\eta}\in(0,1/2)$.

Proof.

To begin with, we translate and rescale $u,V$ to simplify the result. First, we translate the origin such that the flatness hypothesis (3.34) in $B_{\rho}(X^{*})\subset B_{2}$ with

[TABLE]

We dilate the picture by a factor of $2/\rho$. Then it suffices to consider the case $\rho=2$. We set

[TABLE]

which are defined in a ball of radius 2 included in $B_{4/\rho}$. Notice that, if $V\in\mathcal{V}_{2\delta}^{0}$ then $V_{\rho}\in\mathcal{V}_{2\delta}^{0}$.

For simplicity of notation, we drop $\rho$ and use $g_{\rho},V_{\rho}$ to denote $g,V$. After a translation, we have $\eqref{g bounded by V upper and lower}\text{ in some ball }B_{2}(X^{*})\subset\mathbb{R}^{n+1}.$

[TABLE]

It suffices to prove the result (3.35) in a ball $B_{2\bar{\eta}}(X^{*})$.

To prove the desired result, we consider the interior case and boundary case seperately. We distinguish three cases depending on whether $X^{*}$ is close to $\mathcal{L}$, close to $\mathcal{P}$ but away from $\mathcal{L}$, or far from $\mathcal{P}$. Here $\mathcal{P}$ and $\mathcal{L}$ are defined in (2.1).

By Proposition 3.6, we have the equation

[TABLE]

In Case 2 and Case 3 we will need it. Furthermore, $\eta$ below is the universal constant from Proposition 3.11.

Case 1. $|X^{*}|<\eta/4$.

In this case, $B_{1}\subset B_{2}(X^{*})$. The assumptions of Proposition 3.11 are satisfied. We conclude that for any $\bar{\eta}\leq\eta/4$ such that $B_{2\bar{\eta}}(X^{*})\subset B_{\eta}$, either

[TABLE]

or

[TABLE]

Our conclusion holds for all $\bar{\eta}\leq\eta/4$.

Case 2. $|X^{*}|\geq\eta/4$, and $B_{\frac{\eta}{32}}(X^{*})\cap\mathcal{P}=\emptyset$.

As in the proof of Lemma 3.10, by (3.36) we have

[TABLE]

[TABLE]

if $\bar{\epsilon}$ is small enough.

Notice also that by Theorem 7.1 Harnack inequality

[TABLE]

with $C$ universal. Assume that

[TABLE]

Then, in view of (3.36),

[TABLE]

Hence, by Theorem 7.1 Harnack inequality and the condition $C\bar{\delta}^{2}\leq\epsilon$, we have

[TABLE]

Thus, using (3.36) we have that for $\tau$ small enough

[TABLE]

from which our desired conclusion follows with any $\bar{\eta}$ such that $2\bar{\eta}\leq\min\left\{\frac{\eta}{128},\tau\right\}$.

Case 3. $|X^{*}|\geq\eta/4$ and $B_{\frac{\eta}{32}}(X^{*})\cap\mathcal{P}\neq\emptyset$.

The proof is similar to the proof in step 3 of Lemma 3.10. We will construct two functions $q_{1},q_{2}$ to estimate $h$. We will apply Theorem 7.2 boundary Harnack inequality near $\{y=0\}$.

Assume that $X^{*}\in\{y>0\}$ and call $X^{*}_{0}=(x^{*},0)$ the projection of $X^{*}$ onto $\{y=0\}$. Similar to (3.38), for the function $h$ in (3.37) we have

[TABLE]

for a universal constant $C$ if $\bar{\epsilon}$ is small enough.

Denote by $Y^{*}=X_{0}^{*}+\frac{\eta}{16}e_{n}$. We prove the case $u(Y^{*})\geq V(Y^{*}).$ For $u(Y^{*})<V(Y^{*})$, the proof is similar. As in the previous case, by Theorem 7.1 Harnack inequality we have

[TABLE]

Now we argue similarly as in Lemma 3.10.

Denote by

[TABLE]

To estimate $h$, we construct auxiliary functions $q_{1},q_{2}$ satisfying in $D$

[TABLE]

with boundary conditions respectively,

[TABLE]

and

[TABLE]

By the maximum principle, we obtain that

[TABLE]

Moreover,

[TABLE]

Hence, using that $C\bar{\delta}^{2}\leq\epsilon$, we get

[TABLE]

where in the last inequality we used that $U_{n}(Y^{*})q_{1}$ is comparable to $U_{n}$ in view of Theorem 7.2 boundary Harnack inequality.

Finally, by (3.36) and (3.15) we conclude

[TABLE]

Then our desired statement holds for $\bar{\eta}\leq\min\left\{\frac{\tau}{2},\frac{\eta}{64}\right\}$. ∎

Finally, we prove the following corollary. It will be used to indicate the compactness of $u$ during the proof of $C^{2,\alpha}$ regularity.

Corollary 3.13**.**

Let $u$ solve (1.5) and satisfy for $\epsilon\leq\bar{\epsilon}$

[TABLE]

with

[TABLE]

for $\bar{\epsilon},C>0$ universal constants. If $\epsilon\leq\bar{\epsilon}\bar{\eta}^{2m_{0}}$ for some nonnegative integer $m_{0}$ (with $\eta>0$ small universal), then the function

[TABLE]

satisfies

[TABLE]

with

[TABLE]

and $a_{\epsilon},b_{\epsilon}$ having a modulus of continuity bounded by the Hölder function $ot^{\gamma}$, for $o,\gamma$ depending only on $\bar{\eta}$.

Proof.

To begin with, we translate $u$ such that $u$ satisfies (3.34) in $B_{1}$ with $a_{0}=-\epsilon$ and $b_{0}=\epsilon$ for some small $\epsilon\ll\bar{\epsilon}$, and $\delta$ such that $\bar{C}\delta^{2}\leq\epsilon$. By Remark 3.8, the functions $V$ and $g$ are $(4\delta+\epsilon)$-flat in $B_{1}$.

Then, at any point $X^{*}\in B_{1/2}$, we can apply Theorem 3.12 repeatedly for a sequence of radii $\rho_{m}=\frac{1}{2}\bar{\eta}^{m}$ and obtain

[TABLE]

with

[TABLE]

for all $m\geq 1$ such that

[TABLE]

This implies that for all such $m$, the function $\tilde{g}$ satisfies

[TABLE]

with $a_{m}$ and $b_{m}$ as in (3.40).

We then get that in $B_{\frac{1}{4}\bar{\eta}^{m}}(X^{*})\setminus\mathcal{P}$,

[TABLE]

provided that

[TABLE]

If $\epsilon\leq\bar{\epsilon}\frac{\bar{\eta}}{2m_{0}}$ for some nonnegative integer $m_{0}$, then our inequalities above hold for all $m\leq m_{0}$. We thus obtain the following corollary.

∎

4. $C^{2,\alpha}$ Regularity

In this section, we will prove the $C^{2,\alpha}$ regularity by using a sequence of paraboloids to approximate $F(u)$ following the spirit of [DS12] by D. De Silva and O. Savin. To begin with, we consider a sequence of paraboloids approximating the free boundary and analyze the limit. For $\beta=0$, it is easy to analyze since the operator $L_{\beta}=\Delta$. For general $\beta\in(-1,1)$, we will need a more precise analysis.

Lemma 4.1**.**

Suppose there exists a sequence $\lim_{k\to 0}\lambda_{k}=0$ such that

[TABLE]

and

[TABLE]

For $w_{\lambda_{k}}:=\frac{\tilde{u}_{\lambda_{k}}-\gamma_{V_{\lambda_{k}}}}{\lambda_{k}^{1+\alpha}},\;\tilde{u}_{\lambda_{k}}$ defined in Definition 3.1 and $\gamma_{V_{\lambda_{k}}}$ is defined in (3.16), we have that $w_{0}=\lim_{k\to\infty}w_{\lambda_{k}}$ exists and satisfies (2.6).

Proof.

We drop $k$ to simplify the notations. From Proposition 3.7, we have

[TABLE]

The second term tends to [math] as $\lambda$ tends to [math]. By Corollary 3.13, the first term is uniformly Hölder continuous. Furthermore. $w_{0}(0)=0$ and $|w_{0}|\leq 1$.

We show that $w_{0}$ solves (2.6) according to Definition 2.5. We prove in 2 steps. In step 1, we prove that $L_{\beta}(U_{n}w_{0})=0$ in $B_{1/2}\setminus\mathcal{P}$ and show $w_{0}\in C^{1,1}_{loc}(B_{1/2}\setminus\mathcal{P})$. In step 2, we prove the boundary condition $|\nabla_{r}w_{0}|=0$ on $\mathcal{L}$. Here $\mathcal{P}$ and $\mathcal{L}$ are defined in (2.1) and $r:=\text{dist}(X,\mathcal{L})$.

Step 1: $L_{\beta}(U_{n}w_{0})=0$ in $B_{1/2}\setminus\mathcal{P}$ and $w_{0}\in C^{1,1}_{loc}(B_{1/2}\setminus\mathcal{P})$.

Suppose the convergence is $C^{1,t}$ for $t$ close to $1$, it is easy to verify that $L_{\beta}(U_{n}w_{0})=0$ in $B_{1/2}\setminus\mathcal{P}$. In this step, we verify the regularity condition $w\in C^{1,1}_{loc}(B_{1/2}\setminus\mathcal{P})$. We use $u_{\lambda}-V_{\lambda}$ to approximate $U_{t}(\tilde{u}_{\lambda}-\tilde{V}_{\lambda})$ for the desired result. We first prove it is $C^{1,t}$ for a given $t\in(0,1)$, then raise the regularity to $C^{1,1}$ by the property of the equation.

By (4.2), we have that $(\tilde{u}_{\lambda})_{\epsilon,{V}_{\lambda}}$ converges to $w_{0}$ uniformly. As $L_{\beta}u_{\lambda}=0$, by (3.11) we have

[TABLE]

For $d=$dist$(X,\mathcal{P})$, Theorem 7.5 implies

[TABLE]

We will estimate the right hand side to derive the desired regularity. For the first term, we have

[TABLE]

for $\lambda$ small. For $\lambda$ tending to [math], The second inequality is based on that $V_{n}$ converges to $U_{t}(x_{n},y)$ uniformly in $B_{d/2}(X)$.

To estimate the second term, we have $U_{t}(t,y)$ converges to $U_{n}(x_{n},y)$ uniformly in $B_{d/2}(X)$ for $\lambda$ tending to [math]. Therefore $U_{t}$ is uniformly bounded in $B_{d/2}(X)$ for $\lambda$ small. Now we have

[TABLE]

By Arzela-Ascoli theorem, $\frac{u_{\lambda}-V_{\lambda}}{\epsilon}$ converges to a $C^{1,t}_{loc}(B_{1}\setminus\mathcal{P})$ function for $\lambda\to 0$.

Next we raise the regularity to $C^{1,1}$. We use (4.3) to derive that $|L_{\beta}(\frac{u_{\lambda}-V_{\lambda}}{\epsilon})|\leq|y|^{\beta}C_{0}\lambda^{1-\alpha}U_{t}$. Take $\lambda\to 0$, we have that $\frac{u_{\lambda}-V_{\lambda}}{\epsilon}$ converges to a solution to $L_{\beta}u=0$. Invoking Theorem 7.6, the limit is in $C^{1,1}_{loc}(B_{1}\setminus\mathcal{P})$.

Finally, we show that $\frac{u_{\lambda}-V_{\lambda}}{\epsilon}=\frac{{(u_{\lambda}(X)-U)}-{(V_{\lambda}(X)-U)}}{\epsilon U_{n}}U_{n}$ converges to $w_{0}U_{n}$ uniformly in $B_{d/2}(X)$. Since $U_{n}$ is smooth away from the line $\mathcal{L}$, we have $w_{0}\in C^{1,1}_{loc}(B_{d/2}(X))$.

By definition and Taylor expansion, we have

[TABLE]

By Proposition 3.7 and Theorem 7.6, we know that

[TABLE]

in $B_{1}$ for $\lambda$ small. It gives $u_{\lambda}(X)-U=U_{n}(X)\tilde{u}_{\lambda}(X)+O(\lambda^{2})$. Similarly, we have $V_{\lambda}(X)-U(X)=U_{n}(X)\tilde{V}_{\lambda}(X)+O(\lambda^{2})$. Combine these two inequalities together, we have $\frac{u_{\lambda}-V_{\lambda}}{\epsilon}=\frac{{(u_{\lambda}(X)-U)}-{(V_{\lambda}(X)-U)}}{\epsilon}U_{n}=(\tilde{u}_{\lambda})_{\epsilon,{V}_{\lambda}}U_{n}+C\lambda^{1-\alpha}$. Letting $\lambda$ go to [math] and we have the desired result.

Step 2: $|\nabla_{r}w_{0}|=0$ on $B_{1/2}\cap\mathcal{L}$.

In this substep we prove that

[TABLE]

in the viscosity sense of Definition 2.5.

We argue by contradiction. Assume for simplicity (after a translation) that there exists a function $\varphi$ which touches $w_{0}$ from below at [math] with $\varphi(0)=0$ and such that

[TABLE]

Then we can find constants $\sigma$, $\tilde{r}$ small and $A$ large, such that the polynomial

[TABLE]

touches $\varphi$ from below at [math] in a tubular neighborhood $N_{\tilde{r}}=\{|x^{\prime}|\leq\tilde{r},r\leq\tilde{r}\}$ of [math], with

[TABLE]

Therefore we have

[TABLE]

and

[TABLE]

In particular, by continuity near the origin, we can find a point $X^{*}$ such that

[TABLE]

Define

[TABLE]

Then, in view of Proposition 3.7, we have

[TABLE]

Moreover, $W_{\lambda}$ is a subsolution to our problem since $\epsilon\gg\delta^{2}$.

Thus, from the uniform convergence of $w_{\lambda}$ to $w_{0}$ and (4.4), we get that (for all $\lambda$ small)

[TABLE]

Similarly, from the uniform convergence of $w_{\lambda}$ to $w_{0}$ and (4.5), we get that for $k$ large

[TABLE]

On the other hand, it follows from Lemma 3.2 and (4.5) that

[TABLE]

which contradicts (4.6).

∎

Now we prove the following quadratic improvement of flatness proposition. If a solution $g$ stays in a $\lambda^{2+\alpha}$ neighborhood of a function $V\in\mathcal{V}_{1}^{0}$ in a ball $B_{\lambda}$, then in $B_{\eta\lambda}$, $g$ is in a $(\lambda\eta)^{2+\alpha}$ neighborhood of another function $V$ in the same class.

Proposition 4.2**.**

Given $\alpha\in(0,1)$, there exist $\lambda_{0},\eta_{0}\in(0,1)$ and $C>0$ large, depending on $\alpha$, $\beta$, and $n$. If $u$ solves (1.5), $0\in F(u)$, and $u$ satisfies

[TABLE]

for $V=V_{M,0,a,b}\in\mathcal{V}_{1}^{0}$. Then in a possibly different system of coordinates denoted by

[TABLE]

we have

[TABLE]

for some $\bar{V}=\bar{V}_{\bar{M},0,\bar{a},\bar{b}}$ with $\bar{\mathcal{S}}$ given in the $\bar{E}$.

Moreover, for $\bar{V}$ we have

[TABLE]

And the surfaces $\mathcal{S},\bar{\mathcal{S}}$ given by

[TABLE]

separate in $B_{\sigma}$ at most $C(\lambda^{\alpha}\sigma^{2}+\lambda^{1+\alpha}\sigma)$ for any $\sigma\in(0,1]$.

Proof.

We will prove the desired result in by contradiction. Assume that no such $\lambda_{0}$ exists, then, we can find a sequence $\lim_{k\to 0}\lambda_{k}=0$ such that $g_{k}$ and $V_{k}$ satisfying (4.7) and (4.8) fails.

We show the contradiction in two steps. In step 1, we rescale the functions $u$ and $V$ and analyze the limit of function $w_{\lambda_{k}}:=\frac{\tilde{u}_{\lambda_{k}}-\gamma_{V_{\lambda_{k}}}}{\lambda_{k}^{1+\alpha}},\;\tilde{u}_{\lambda_{k}}$ defined in Definition 3.1 and $\gamma_{V_{\lambda_{k}}}$ is defined in (3.16). In step 2, we apply Theorem 2.6 to construct $\bar{V}$ to derive the contradiction.

Step 1: Rescaling and $w_{\lambda}$.

To begin with, we rescale $g_{k}$ and $V_{k}$. For simplicity of notation, we drop the dependence on $k$ and denote

[TABLE]

Now we have

[TABLE]

From Lemma 4.1, we have that $w_{0}=\lim_{k\to\infty}w_{\lambda_{k}}$ is a solution to (2.6)

Step 2: Construction of $V$.

In this step, we construct a new function $\bar{V}$ such that $\tilde{g}_{\lambda}$ can be approximated by $\bar{V}$ for $\lambda$ small. We will use the uniform convergence of $w_{\lambda}$ to $w_{0}$ to construct a polynomial approximating it. By Proposition 3.7 we can construct a solution related to the polynomial. After a modification of the coordinate system, we have constructed $\bar{V}$.

In step 2, we have proven that $w_{0}$ is a solution to (2.6). By Theorem 2.6, we have

[TABLE]

Then for $\eta_{0}$ small enough depending on $\alpha$, we have

[TABLE]

Set $T_{\lambda}:=V_{\lambda M-\epsilon M_{0},\epsilon\xi_{0},\lambda a-\epsilon a_{0},\lambda b-\epsilon b_{0}}$. Here we dropped $k$ to simplify the notation. By Proposition 3.7, we have

[TABLE]

From the uniform convergence of $w_{\lambda}$ to $w_{0}$, t follows that for fixed $r$ and sufficiently large $k$, $\lambda_{k}$ satisfies

[TABLE]

By the definition of $w_{\lambda}$, we have

[TABLE]

Rescale $u_{\lambda}$ back from $B_{1}$ to $B_{\lambda}$ and obtain

[TABLE]

Finally, we show that $\mathcal{S}$ and $\bar{\mathcal{S}}$ separate. Then (4.8) follows by definition. We will consider the function $T=V_{M_{T},0,a_{T},b_{T}}$ and the surface $\mathcal{S}_{T}:=\{x_{n}:=\frac{1}{2}(x^{\prime})^{T}M_{T}x^{\prime}+\xi_{T}\cdot x^{\prime}\},$ with

[TABLE]

The distance between $\mathcal{S}$ and $\mathcal{S}_{T}$ can be derived by simple calcuius. It suffices to show that $\mathcal{S}_{T}$ approximates $\bar{\mathcal{S}}$.

To begin with, we construct a different system of coordinates $\bar{E}$. Without loss of generalization, we assume $\xi_{0}$ is in $e_{1}$ direction. We choose $\bar{E}={\bar{e}_{1},\cdots,\bar{e}_{n+1}}$ with $\bar{e}_{i}=e_{i}$ for $i\neq 1,n$, and $\bar{e}_{n}$ normal to $\epsilon\xi_{0}\cdot x^{\prime}$ at [math]. $\bar{E}$ is obtained by an orthogonal tranformation $O$ such that $\bar{x}=Ox$. Furthermore, we have that $\|O-I\|\leq C\epsilon$.

By definition, we have that

[TABLE]

In the original system $E$, there exists a function $g$ such that

[TABLE]

By the property of $O$, we have that

[TABLE]

Then in $B_{2\eta_{0}\lambda}$ we have that the surfaces $\mathcal{S}_{T}\pm\frac{\epsilon}{2}\lambda\eta_{0}^{2+\alpha}e_{n}$ lie between $\bar{\mathcal{S}}\pm\epsilon\lambda\eta_{0}^{2+\alpha}\bar{e}_{n}$. We are done.

Now we prove (4.8). We set $\bar{V}=V_{M_{T},0,a_{T},b_{T}}$ in $\bar{E}$. By the analysis above, we have the distance between $\bar{\mathcal{S}}$ and $\mathcal{S}_{T}$. Together with (4.10), we have the desired result. ∎

Finally, we prove that flat free boundaries are $C^{2,\alpha}$ to end this section.

Theorem 4.3**.**

For $\alpha\in(0,1)$, there exists $\bar{\epsilon}$ small depending on universal constants and $\alpha$, such that if $u$ solves (1.5) in $B_{1}$ with

[TABLE]

then $F(u)\cap B_{1/2}^{n}$ is $C^{2,\alpha}$.

Proof.

Our goal is to construct a paraboloid that a paraboloid approximates $F(u)$ with $C^{2,\alpha}$ precision at the origin. The argument can then be applied to any point in $F(u)\cap B_{1/2}$. The proof proceeds as follows. First we apply Proposition 4.2 iteratively to generate a sequence of approximating functions $V_{k}$ and coordinate systems $E_{k}$. Second, we show that the corresponding surfaces $\mathcal{S}_{k}$ converge uniformly to a limit paraboloid $\mathcal{S}$, which provides the desired approximation for $F(u)$.

To begin with, we consider the rescaling $u_{\mu}(X)=\mu^{-s}u(\mu X)$. By Lemma 3.3, we have

[TABLE]

provided that $\bar{\epsilon},\mu$ small depending on $\tau\leq\lambda_{0}$. Here $\lambda_{0}$ is from Proposition 4.2 and $\tau$ is to be determined later.

Applying Proposition 4.2 repeatedly to $u_{\mu}$, we have a sequence $\tau_{k}=\tau\eta_{0}^{k}$, $V_{\mathcal{S}_{k},a_{k},b_{k}}$, and vector $e_{n}^{k}$ such that

[TABLE]

We choose $\tau$ small such that $\sum C\tau_{k}^{\alpha}\leq 1$, therefore $M_{k}\to M_{*}$, $a_{k}\to a_{*}$, $b_{k}\to b_{*}$ always have norm no more than $1$. Furthermore, we have

[TABLE]

Then $F(u_{\mu})$ is $C^{2,\alpha}$ at [math].

From Proposition 4.2, $\mathcal{S}_{k}$ and $\mathcal{S}_{k+1}$ separate in $B_{\sigma}$ at most $C(\tau_{k}^{\alpha}\sigma^{2}+\tau_{k}^{1+\alpha}\sigma)$, $\mathcal{S}_{k}$ converge uniformly to a paraboloid $\mathcal{S}_{*}$ in $B_{1}$. Since $\mathcal{S}_{k}$ and $\mathcal{S}_{*}$ separate in $B_{2\tau_{k}}$ at most $C\tau_{k}^{2+\alpha}$, we apply Lemma 3.9 to derive that

[TABLE]

Then $F(u_{\mu})$ is $C^{2,\alpha}$ at [math]. ∎

5. $C^{\infty}$ Regularity

In this section, we provide the proof of Theorem 1.3. From Theorem 4, we have that the free boundary is $C^{2,\alpha}$. A recent work [BROW25] by B. Barrios, X. Ros-Oton, and M. Weidner provided a different proof for this step by considering a general type of global elliptic operator. We will prove using a different method.

In this section, we will show that the derivatives $g_{i}$ has the same regularity as $g$ as in [DS15] by D. De Silva and O. Savin. In the following theorem, we consider the behavior of $\nabla_{x}^{2}u$ near the free boundary. Without loss of generality, we consider the following case throughout this section.

[TABLE]

Theorem 5.1**.**

If $u$ is a solution to (1.5) with $F(u)$ satisfying (5.1) for $k\geq 0$, then we have:

[TABLE]

[TABLE]

[TABLE]

where $U_{0}$ is defined in (3.1) and $C$ is a universal constant.

Proof.

Note that (5.2) and (5.3) follow from [JN17, Proposition 3.2 and Lemma 7.2]. It suffices prove that (5.4). Without loss of generalization, we assume that the normal unit of $F(u)$ at [math] is $e_{n}$.

By (5.2), there exists a degree $k+1$ polynomial $P(x,r)$ such that $|u-U_{0}P|\leq CU_{0}r^{k+1+\alpha}$. We claim that for any $1\leq i,j\leq n$ we have

[TABLE]

Then for each $x\in F(u)$, we apply (5.5) to

[TABLE]

As $\bigcup_{x\in F(u)}\mathcal{K}_{x}$ is the whole space, we have the desired result.

We prove (5.5) by scaling. To begin with, we define the following scalings for $\lambda\in(0,1)$:

[TABLE]

By choosing different $\lambda$, $\mathcal{K}\cap(B_{3\lambda/4}\setminus\bar{B}_{\lambda/4})$ covers $\mathcal{K}$. Therefore it suffices to show that

[TABLE]

We will prove (5.6) in 3 steps. In step 1, we determine the equation satisfied by $\tilde{U}_{\lambda}$. In step 2 and 3, we prove the desired result for $x_{n}\geq 0$ and $x_{n}<0$ using Theorem 7.1 Harnack inequality and Theorem 7.2 boundary Harnack inequality.

Step 1: equations for $\tilde{U}$.

By (5.2), $\|\tilde{U}_{\lambda}\|_{L^{\infty}(B_{1})}\leq C$. First we consider $L_{\beta}\tilde{U}_{\lambda}$. Following [JN17, Theorem 4.1], we claim that

[TABLE]

Here $F(X)=-\frac{U_{0,\lambda}}{r_{\lambda}}\frac{h(\lambda X)}{\lambda^{k+s}}$, $h(x,y)=\sum_{l=0}^{k}r^{l}h_{l}(x)$, $\|h_{l}\|_{C^{k,\alpha}(B_{1}^{n})}\leq C(n,\beta)$ such that $h_{l}$ has vanishing derivatives up to order $k-l$ at zero. We rewrite the equation of $u$ to be

[TABLE]

We prove the claim to end this step. It suffices to prove for the case $\lambda=1$. The result for general $\lambda$ follows by scaling. As $L_{\beta}u=0$, we only need to consider $L_{\beta}(U_{0}P)$. By calculation following [JN17], we have

[TABLE]

where $\kappa$ is defined in (3.8). Here $c_{\sigma l}^{\mu m}\neq 0$ only if $|\mu|+m\leq|\sigma|+l\leq k$, and

[TABLE]

h^{\mu}_{m},h^{\mu}_{m-1}\in C^{k,\alpha}(B_{1}^{n})\text{ has vanishing derivatives of order k-m$and$k-m+1 at zero.}

Step 2: (5.6) in $\{x_{n}\geq 0\}$.

Since $\|F(u)\|_{C^{k+2}}\leq 1$, $U_{0,\lambda}$ has a positive lower bound in $\mathcal{K}\cap(B_{3/4}\setminus\bar{B}_{1/4})\cap\{x_{n}\geq 0\}$. We only need to show that $\nabla_{x}^{2}\tilde{U}$ is uniformly bounded.

We claim that the right hand side of (5.7) is $C^{\alpha}$ bounded and apply Schauder estimate to (5.7). The desired result will follow.

The regularity of $F$ is natural. We focus on the regularity of $\frac{\beta}{y}\partial_{y}\tilde{U}+\partial_{yy}\tilde{U}$. By calculus, we have

[TABLE]

The transformation was pointed out in [CS07]. We apply Prop. 7.4 to the equation, it gives

[TABLE]

Similarly, we have that

[TABLE]

We apply Proposition 7.4 again to derive that

[TABLE]

We apply the same trick for the third time to derive

[TABLE]

As $|y|^{\beta}\partial_{y}(|y|^{-\beta}\partial_{y}(|y|^{\beta}\partial_{y}\tilde{U}))=|y|^{\beta}\partial_{y}(\frac{\beta}{y}\partial_{y}\tilde{U}+\partial_{yy}\tilde{U})$, the inequality gives the Hölder continuity of $\frac{\beta}{y}\partial_{y}\tilde{U}+\partial_{yy}\tilde{U}$ in the $y$ direction. For the Hölder continuity in $x$-direction, we consider the equation

[TABLE]

We apply Prop. 7.4 to the equation, it gives that

[TABLE]

Step 3: (5.6) in $\{x_{n}<0\}$.

We notice that, if we can prove the inequlity (5.6) near $\{y=0\}$, then we only need to check boundness for points away from $\{y=0\}$. That was proven in Step 2.

As $\partial_{ij}\tilde{U}=0$ on $\{y=0\}$, it suffices to show that $|y|^{\beta}\partial_{y}(\partial_{ij}\tilde{U})$ is bounded. We consider

[TABLE]

Apply Proposition 7.4 to the equation, we have

[TABLE]

∎

Now we try to improve the regularity of $F(u)$. We want to show that the derivatives of the free boundary has the same regularity as itself. To prove that, we need to analyze the PDE of it. We notice that $-\frac{u_{i}}{u_{n}}$, $i\leq n-1$ is a natural extension of the derivative. It satisfies $L_{\beta}(u_{n}\frac{u_{i}}{u_{n}})=0$. To analyze the regularity of $\frac{u_{i}}{u_{n}}$, we state the following theorem.

Theorem 5.2**.**

Let $u$ be a solution to (1.5) and $\|F(u)\|_{C^{k+2,\alpha}(B_{1})}\leq\frac{1}{2}$. Assume $w\in C(B_{1})$, even in $y$, satisfies the equation

[TABLE]

Then $\|w\|_{C^{k+2,\alpha}_{xr}(F(u)\cap B_{1/2})}\leq C$ with $C$ depending only on $n,k,\alpha.$

The case $F(u)=\mathcal{L}$ in (2.1) is Theorem 2.6. For the general case, we will use iteration to construct a sequence of $C^{k+2}$ functions approximating $w$. To begin with, we follow [DS15, Section 7.1] to define $E(Q)$. For $z=(z^{\prime},z_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}$, we set

[TABLE]

Since $\nu$ has lower regularity, $T_{0}$ only has $C^{k+1,\alpha}$ regularity. However, the restriction of $T$ on the hyperline $\mathcal{L}$ in 2.1 is pointwise $C^{k+2,\alpha}$. Furthermore, $T$ is one-to-one near the origin. Then we define

[TABLE]

By the analysis above, $Q$ is pointwise $C^{k+2,\alpha}$ on $F(u)$. By Whitney extension, we have

Theorem 5.3**.**

([DS15, Thm. 7.1] Whitney Extension Theorem) There exists $E(Q)$ such that

[TABLE]

and

[TABLE]

Moreover, $E(Q)$ is linear in $Q$. For $Q$ in (5.9), we have

[TABLE]

We can consider $E(Q)$ as $Q$ in Theorem 2.6. We want to have a polynomial type function to approximate $u$. Furthermore, the property of $Q$ in Theorem 2.6 implies that we need a function only depending on the projection onto $\Gamma$. However , that is impossible due to the regularity. Therefore we use Whitney extension to construct $E(Q)$. It extends $Q$ near $\Gamma$ in $C^{k+2.\alpha}$ regularity. Now we consider the equation for $E(Q)$.

Lemma 5.4**.**

Let $e$ be a unit vector and $u_{e}$ has the following expansion at [math],

[TABLE]

Then for polynomial $Q$ such that

[TABLE]

we have

[TABLE]

where $R$ is a polynomial of degree $k$ in $(x,r)$ given by

[TABLE]

Here $A_{\sigma l}$ satisfies

[TABLE]

Additionally,

[TABLE]

Proof.

Since $E(Q)$ is independent of $y$ direction and $L_{\beta}u_{e}=0$, $L_{\beta}(u_{e}E(Q))$ only have two main terms:

[TABLE]

We will now find the expansion for each of these terms at the origin.

First, we analyze the term involving $\Delta E(Q)$. By Theorem 5.3, we know it is pointwise $C^{k,\alpha}_{x}$ at the origin. We formally differentiate it at the origin to find the expansion. We have

[TABLE]

For the second term, we claim that

[TABLE]

for two bounded vectors $\xi,\eta\in\mathbb{R}^{n}$ with $\xi\cdot\nu=0$.

Assume for simplicity that $x$ is a point on the $e_{n}$-axis. Then, since $E(Q)\in C^{k+2,\alpha}_{x}$, we find

[TABLE]

From Taylor expansion, we see that

[TABLE]

for some bounded vector $\xi$. Moreover, since $Q$ is constant on perpendicular lines to $\alpha$, we have $\nabla Q(x)\cdot e_{n}=0$ and $\partial_{n}^{l}Q(0)=0$ for all $l\leq k+2$. Thus, the formula above gives $\xi\cdot e_{n}=0$, and our claim is proved.

Next we give the estimate of $u_{ei}$. By Theorem 5.1, we have that

[TABLE]

Therefore we have that

[TABLE]

Finally, we derive the estimate of $L_{\beta}(u_{e}E(Q))$.

[TABLE]

To obtain the desired result, we only need to apply the expansion of $P_{0}^{e}$ and $\nabla P^{e}_{0}$ at [math] to the equality above. ∎

Next we prove the compactness of solutions to (2.6). Suppose that $u$ is a solution to (1.5), by Theorem 5.1, we have

[TABLE]

We consider the scaling

[TABLE]

it satisfies the expansion

[TABLE]

We notice that we only care about the regularity of $w$ near the free boundary $F(u)$. Therefore we can always scale $u$ and $w$. In the following Theorem, we only consider the scaled case.

Lemma 5.5**.**

Suppose $u$ is a solution to (1.5) and satisfies the scaled expansion (5.10) for some $\delta$ small enough. For $w$ satisfying

[TABLE]

We have $w\in C^{\eta}$ and $\|w\|_{C^{\eta}(B_{1/2})}\leq C,$ for some small, universal $\eta$.

Proof.

First, we note that for $\delta$ small enough, the equation is non-degenerate away from $F(u)$. Therefore the standard elliptic estimates guarantee the interior regularity of $w$.Our main task is to prove Hölder continuity near the set $\{u=0\}$. We handle this in two distinct regions. In step 1, we prove that $w$ is Hölder near $F(u)$. In step 2, we prove that $w$ is Hölder near $\{u=0\}\setminus F(u)$.

Step 1: Regularity on $F(u)$.

In this step, we use the iteration argument to prove the Hölder regularity of $w$ near $F(u)$. Under the setting (5.1), we claim that

[TABLE]

for some $\delta_{0}$ universal. We notice that after a translation $w_{t}(X)=w(X)+t$, we can always consider the case $\text{osc}_{B_{1}}w=2\|w\|_{L^{\infty}}$. Suppose the claim holds, we have that

[TABLE]

By iteration, we have that $\|w\|_{C^{\eta_{1}}(\alpha)}\leq C$ for some $\eta_{1}\in(0,1)$.

We prove the claim in 2 substeps. In substep 1, we construct barrier function $v$. In substep 2, we compare the barrier function $v$ with $u$. We will construct lower barrier function $v$ under the setting $w(\frac{1}{2}e_{n})\geq 0$. For the case $w(\frac{1}{2}e_{n})\leq 0$, we can construct the upper barrier function similarly.

Substep 1: Barrier function $v$.

We set

[TABLE]

where $Q(y^{\prime})=-|y^{\prime}|^{2}$, $M$ large and $\delta_{1}$ small to be determined later. By Lemma 5.4, we have

[TABLE]

By calculation we have

[TABLE]

Here $\kappa$ is defined in (3.8). We choose $M$ big enough and $\delta$ small enough such that $L_{\beta}(U_{0}(1+Mr))\geq 2C|y|^{\beta}\frac{U_{0}}{r}$. By the two inequalities above, we have

[TABLE]

On $F(u)$ we have

[TABLE]

Substep 2: Comparison of $w$ and $v$ in $B_{3/4}\cap\{r<c\}$.

In this substep, we compare $w$ and $v$ in $B_{3/4}\cap\{r<c\}$. We start from compare them on $\partial B_{3/4}\cap\{r<c\}$. For $c,\delta$ small enough, we have

[TABLE]

Therefore we have $v<-1\leq w\text{ on }\partial B_{3/4}\cap\{r<c\}$.

Next we compare $w$ and $v$ on $B_{3/4}\cap\{r=c\}$. We claim that that $1+w\geq c_{0}>0$ for some universal $c_{0}>0$ on $B_{3/4}\cap\{r=c\}$. By choosing $c,\delta_{1}$ small enough, we have $w\geq-1+c_{0}\geq v$ on $B_{3/4}\cap\{r=c\}$.

We use Theorem 7.1 Harnack and Theorem 7.2 boundary Harnack inequalities to prove the claim. Since $1+w(\frac{1}{2}e_{n})\geq 1$ and $\delta$ is small, we apply Harnack inequality to compare $u_{n}(1+w)$ with $u_{n}$. It gives that

[TABLE]

Similarly, we apply Theorem 7.2 boundary Harnack inequality to compare $u_{n}(1+w)$ with $u_{n}$ under the setting $\delta$ small enough. The comparison is extended to the boundary $\{u=0\}$ and the claim follows.

Finally, we show $w\geq v$ in $B_{3/4}\cap\{r<c\}$. For $\delta<<\delta_{1}$, (5.11) indicates that the minimum of $u_{n}(w-v)$ cannot occur inside by maximum principle. The free boundary condition indicates that the minimum of $u_{n}(w-v)$ cannot occur on $F(u)$. As analyzed above, the minimum of $u_{n}(w-v)$ cannot occur on $\partial(B_{3/4}\cap\{r<c\})$. That shows $v$ is a lower barrier function.

Step 2: Regularity on $\{u=0\}\setminus F(u)$.

For $X\in\{u=0\}\setminus F(u)$ and $2R=dist(X,F(u))\leq 1$, we prove the Hölder continuity of $w$ in $B_{R}(X)$. We prove it in 2 substeps. In substep 1, we set $H=u_{n}w$ and show the Hölder continuity of $\frac{H}{|y|^{1-\beta}}$. In substep 2, we show the Hölder continuity of $\frac{|y|^{1-\beta}}{u_{n}}$. The Hölder continuity of $w$ will be derived by the fact that $w=\frac{H}{|y|^{1-\beta}}\cdot\frac{|y|^{1-\beta}}{u_{n}}$.

Substep 1: Hölder continuity of $\frac{H}{|y|^{1-\beta}}$.

In this substep, we show the Hölder continuity of $\frac{H}{|y|^{1-\beta}}$, $H=u_{n}w$. Set $\tilde{H}(X)=H(RX)$, we have

[TABLE]

By calculus we have

[TABLE]

Apply Theorem 7.5, we have

[TABLE]

Scaling it back, we have

[TABLE]

Substep 2: Hölder continuity of $\frac{|y|^{1-\beta}}{u_{n}}$.

We use Theorem 7.2 boundary Harnack inequality to show the Hölder continuity of $\frac{|y|^{1-\beta}}{u_{n}}$. As we have

[TABLE]

Apply the boundary Harnack theorem, we have

[TABLE]

The desired result follows. ∎

With the uniform Hölder estimate, we can derive a compactness result by Arzela-Ascoli theorem. Here we give the result as below.

Lemma 5.6**.**

Assume a sequence $u_{k}$ of solutions to (1.5) satisfying (5.10) for a sequence $\delta_{k}\to 0$. If a sequence $w_{k}$ satisfies

[TABLE]

then there is a subsequence of $w_{n_{k}}$ of $w_{k}$ such that $w_{n_{k}}$ converges uniformly on compact sets to a Hölder continuous function $w$. Furthermore, $w$ satisfies (2.6).

Proof.

By Lemma 5.5, the existence and continuity of the limit $w$ is a direct result of Arzela-Ascoli Theorem. It suffices to prove that $w$ satisfies (2.6).

For the interior part, Theorem 7.5 indicates that $w$ is $C^{1,t}$ for any $t<1$. Condition (5.10) indicates that $F(u)\to\mathcal{L}$, $(u_{k})_{n}\to(U_{0})_{n}$. Therefore $w$ satisfies $L_{\beta}(U_{n}w)=0$ in $B_{1}\setminus\mathcal{P}$.

For the boundary condition, we prove by contradiction. Suppose $w$ can be touched from below strictly at [math] by a function $\varphi$ such that

[TABLE]

for some $b_{0}$ and $y_{1}^{\prime}$. Since $w_{k}\to w$ uniformly, then we can touch $w_{k}$ from below by $\varphi_{k}=b_{k}-2a_{1}E(|y^{\prime}-y^{\prime}_{1}|^{2})+s\frac{U_{0}}{u_{n}}(\frac{a_{2}}{2}+Mr)$.

Now, as in Step 1 in the proof of Lemma 5.5, we can show that $\varphi_{k}$ is a subsolution to (2.6) for $k$ big enough. Therefore it cannout touch $w_{k}$ from below. This gives a contradiction. ∎

With all the preparation above, we give the proof of Theorem 5.2 below.

Proof.

In the following proof, we will show that $w$ is $C^{k+2,\alpha}$ at [math]. The proof applies to all points on $F(u)$. We notice that $C^{k+2,\alpha}$ is a local regularity, therefore it suffices to consider a scaling of $w(RX)$ for $X\in B_{1/R}$. After a scaling, we only need to consider the case (5.10) for some $\delta$ small.

By definition, it suffices to find a $C^{k+2}$ function $W_{Q,P}$ to approximate $w$ at [math]. We determine $W_{Q,P}$ by approximating $w$ in sequence of balls $B_{\rho^{m}}$ inductively on $m$. Set

[TABLE]

with

[TABLE]

We claim that: if for a given $W_{Q,P}$ such that $\|w-W_{Q,P}\|_{L^{\infty}(B_{1})}\leq 1$ and $Q,P$ satisfy

[TABLE]

then there exists some $Q^{\prime},P^{\prime}$ satisfying (5.14) such that $\|(Q^{\prime}+rP^{\prime})-(Q+rP)\|_{L^{\infty}(B_{1})}\leq C$, and $\|w-W_{Q^{\prime},P^{\prime}}\|_{L^{\infty}(B_{\rho})}\leq\rho^{k+2+s}$ for $\rho$ universal.

We notice that the above claim implies that there exists $Q_{m},P_{m}$ such that $W_{Q_{m},P_{m}}$ approximating $w$ in $B_{\rho^{m}}$. We set $\bar{Q}=\lim_{m\to\infty}Q_{m},\bar{P}=\lim_{m\to\infty}P_{m}$, then $|w-W_{\bar{Q},\bar{P}}|\leq C|X|^{k+2+s}$ by standard argument. That is the desired result.

Now it suffices to prove the claim. To begin with, we start by investigating (5.14) $(i)$. It indicates that $\partial_{\nu}W_{Q,P}=O(|X|^{k+1+s})$. By Lemma 5.1, we have $\frac{U_{0}}{u_{n}}=r(s+P^{*})$. Therefore it suffices to require $P^{*}P$ to vanish of order $k+1$ at origin. Consider the coefficients of $x^{\mu^{\prime}}$, we have

[TABLE]

Next we analyze (5.14) $(ii)$. We calculate as below

[TABLE]

where A_{\sigma l}\text{ is given in Lemma \ref{equation for u_nEQ}}. For the second term above, we calculate following [JN17] (4.1) to derive that

[TABLE]

Here $c_{\sigma l}^{\mu m}\neq 0\text{ only if }|\mu|+m\leq|\sigma|+l\leq k,$ $|c_{\sigma l}^{\mu m}|<\delta$, and $h^{\mu m}(x,r)=r^{m}h_{m}^{\mu}(x)+mr^{m-1}h_{m-1}^{\mu}(x)$ and $h^{\mu}_{m},h^{\mu}_{m-1}\in C^{k,\alpha}(B_{1}^{n})$ has vanishing derives of order $k-m$ and $k-m+1$ at the origin. That indicates

[TABLE]

where $B_{\sigma l}$,$h_{m}$ satisfy

[TABLE]

In conclusion, (5.14) $(ii)$ is equal to $A_{\sigma l}+B_{\sigma l}=0$. Now we show that $q_{\mu}$ and $a_{\mu 0}$ in (5.13) will uniquely determine $Q$, $P$ satisfying (5.14). We prove by induction. Suppose for $l\in\mathbb{N}$, $a_{\mu l}$ are determined. Then we have

[TABLE]

As $A_{\sigma l}$ is uniquely determined by $q_{\mu}$ and $c_{\sigma l}^{\mu m}\neq 0\text{ only if }|\mu|+m\leq|\sigma|+l\leq k,$, we are done. Furthermore, by (5.15) we only need $q_{\mu}$ and $a_{\mu,0}$, $\mu\neq 0$ to determine $Q,P$ uniquely.

We prove the claim to end this proof. Set

[TABLE]

we have

[TABLE]

From Lemma 5.6 and Theorem 2.6, for $\delta$ small enough there exists $\tilde{Q}$ and $\tilde{P}$ such that

[TABLE]

Furthermore, $\tilde{Q}$ and $\tilde{P}$ satisfy (5.14) for $F(u)=\{(x^{\prime},x_{n},y):x_{n}=y=0\}$. We modify $\tilde{Q},\tilde{P}$ to construct $Q^{\prime}$, $P^{\prime}$ in the claim.

By the argument above, we use $q_{\mu}$ of $\tilde{Q}$ and $\tilde{a}_{\mu 0}$ of $\tilde{P}$ as coefficients. The generated $Q^{\prime}$,$P^{\prime}$ satisfying (5.14) is unique. Furthermore, since $\tilde{Q}$ and $\tilde{P}$ satisfy the equations (5.15) and (5.16) with $c_{\sigma l}^{\mu m}=\hat{c}^{\mu}_{\sigma^{\prime}}=0$, we have $\|\tilde{Q}+r\tilde{P}-\bar{Q}-r\bar{P}\|\leq C\delta$. Finally, by the fact that $E(\bar{Q})$ is close to $\bar{Q}$ for $\rho$ small, we have

[TABLE]

That gives the desired claim. ∎

Finally, we prove Theorem 1.3 to end this section.

Proof.

Since $u(x^{\prime},g(x^{\prime}),0)=0$, simple calculus gives that

[TABLE]

Therefore it suffices to show that $w=\frac{u_{i}}{u_{n}}$ has the same regularity as $g$. We prove in 2 steps. In step 1, we apply Theorem 5.1 to show that $\frac{u_{i}}{u_{n}}$ satisfies the equation (5.8). In step 2, we apply Theorem 5.2 to show that $w$ has the same regularity as $g$.

Step 1: Equation for $w$.

For the interior equation, we have

[TABLE]

it suffices to prove the Neumann boundary condition.

From Theorem 5.1 and $\nabla g(0)=0$, there exists a polynomial $P=a_{0}+a_{n}x_{n}+a_{n+1}r$, $a_{n}\neq 0$ such that $\frac{u_{i}}{sU_{0}/r}=\partial_{i}P+O(r^{1+\alpha})$, $1\leq i\leq n$. Therefore we have

[TABLE]

That indicates the Neumann condition.

Step 2: Improvement of regularity.

From Theorem 5.2, $w$ has the same regularity as $g$. Suppose $g$ is $C^{k+2,\alpha}$, then $g_{i}$ is also $C^{k+2,\alpha}$. That implies $g$ is $C^{k+3,\alpha}$. The induction shows that $g$ is smooth. ∎

6. Expansion Theorem for Linearized Problem

In this section, we prove the expansion Theorem 2.6. It gives an infinite order expansion to the linearized problem (2.6). For $\beta=0$, the equation $\Delta(U_{n}w)=0$ in $\mathbb{R}^{n}\setminus\mathcal{P}$ is easier to understand. We can consider the slice of the space $\{x^{\prime}=0\}$ and map the 2D slit domain to the upper half plane with a holomorphic map. For general $\beta$, we will use boundary Harnack inequality in slit domain to handle it. The idea is inspired by [DS12, JN17].

For $k=1$, the theorem was proven in [DS16] by D. De Silva, O. Savin, and Y. Sire. Based on that, we will prove by induction on $k$. Our proof is inspired by the paper [DS12] by D. De Silva and O. Savin. Furthermore, the regularity results in [JN17] by Y. Jhaveri and R. Neumayer and [STV21a] by Y. Sire, S. Terracini, and S. Vita play important roles in our proof. To begin with, we recall the boundary Harnack theorem in slit domain. It is a special case of Theorem 7.3.

Theorem 6.1**.**

([JN17, Proposition 3.3]) Assume that $u\in C(B_{1})$ is a solution to (1.5), even with respect to $\{y=0\}$, and $F(u)=\mathcal{L}$. For any $k\in\mathbb{N}$, there exists a polynomial $P(x,r)$ of degree $k$ with $\|P\|\leq C$ such that

[TABLE]

for some constant $C=C(\beta,n,k)$.

Remark 6.2**.**

Notice that for $k\in\mathbb{N}$, it suffices to have $|L_{\beta}u|\leq C|y|^{\beta}|X|^{k-1}U$ to derive the same result. To prove this, we calculate as below.

[TABLE]

By the calculation above, for $u$ satisfying $|L_{\beta}u|\leq C|y|^{\beta}|X|^{k-1}U$, it suffices to consider $u\pm C|X|^{k+1}U$ as barrier functions. In detail, we have $L_{\beta}(u+C|X|^{k+1}U)\geq 0$. Then we consider $u_{1}$ satifying

[TABLE]

By Theorem 7.1 Harnack inequality, we have $|u_{1}-UP_{1}|\leq C|X|^{k+1}U$. By maximum principle, we have

[TABLE]

Similarly, we can derive that

[TABLE]

We can simply derive that

[TABLE]

As $P_{1}$ and $P_{2}$ are polynomials of degree $k$, we can say they are exactly the same.

The proof of Theorem 2.6 is based on the fact that for the solution $h$ to (2.6), $D^{k}_{x^{\prime}}h$ has the same regularity as $h$. To begin with, we present some known results first.

Lemma 6.3**.**

([DSSS14, Theorem 6.1 and Lemma 6.3]) Given a boundary data $\bar{h}\in C(\partial B_{1})$ even with respect to $\{y=0\}$, $|\bar{h}|\leq 1$. There exists a unique classical solution $h$ to (2.6) such that $h\in C(\bar{B}_{1})$ even with respect to $\{y=0\}$, $h=\bar{h}$ on $\partial B_{1}$.

Furthermore, $D^{k}_{x^{\prime}}h$ is also a solution to (2.6) for any $k\in\mathbb{N}^{n}$. There exists $\alpha\in(0,1)$ universal such that

[TABLE]

Here $|\xi_{1}|\leq C(n,\beta)$.

Remark 6.4**.**

Lemma 6.3 indicates the Hölder regularity of $D^{k}_{x^{\prime}}h$. In fact, $D^{k}_{x^{\prime}}h$ shares the same regularity with $h$. By Lemma 6.3 we have that $D^{k}_{x^{\prime}}h$ is also a solution to (2.6) with uniform bound. We apply the regularity estimate of $h$ to $D^{k}_{x^{\prime}}h$ and have the result.

Now we give the proof of Theorem 2.6. Our proof is inspired by the paper [DSSS14].

Proof.

We will prove the desired results in 2 steps. In step 1, we show that for $\alpha$ in Lemma 6.3 and $\forall k\in\mathbb{N}$, there exists a polynomial $P_{k}(x,r)$ of order $k$ such that

[TABLE]

In step 2, we consider the properties of $P_{k}$. We show that $P_{k}$ is bounded and $L_{\beta}(U_{n}P_{k})=0$ for all $k$. Then by the bound of $P_{k+1}$, we have that the $\alpha$ in step 1 can be raised to 1.

Step 1: Proof of (6.1):

We prove by induction on k in this step. The case $k=1$ was proven in [DSSS14, Theorem 6.1]. We assume (6.1) for $k$ and prove the $k+1$ case. We will prove in 2 substeps. In substep 1, we show that there exists a polynomial approximating $h$ in the space $\{x^{\prime}=0\}$. In substep 2, we extend the result to the whole space.

Substep 1: Polynomial appromating $h$ in $\{x^{\prime}=0\}$.

In this step, we consider $h$ in $\{x^{\prime}=0\}$. In this 2D space, we will use coordinates $t$ to denote original $x_{n}$. $y,r$ remain the same as in $\mathbb{R}^{n+1}$. By definition, we have

[TABLE]

Since $U_{n}$ and $y$ are independent of $x^{\prime}$, we extract the terms of derivatives in $x^{\prime}$ direction to have

[TABLE]

As indicated in Remark 6.4, we can apply the regularity result of $h$ to $f=\Delta_{x^{\prime}}h$ to derive

[TABLE]

where $C_{o}$ is a universal constant and $\|Q\|$ is bounded by universal constants and $k$. Here $Q$ is an order $k$ polynomial of $t,r$. In $\{x^{\prime}=0\}$, $\Delta_{x^{\prime}}h$ is a function of $t$ and $r$. We use $f$ to denote $\Delta_{x^{\prime}}h$. We investigate the 2-dimensional problem in $(t,y)$-variables:

[TABLE]

The regularity analysis of (6.2) derives regularity results on sets $\{x^{\prime}=0\}$. We consider the solution $H\in C(B_{1/2})$ to the following problem

[TABLE]

We claim that $H=U_{t}h$ in $B_{1/2}$. By the equality, it suffices to analyze the regularity of $H$. By Remark 6.4, we have that for some universally bounded polynomial $P(t,r)$ of order $k+1$,

[TABLE]

This gives the desired result in the plane $\{x^{\prime}=0\}$.

We prove the claim to end this substep. As we have

[TABLE]

it suffices to analyze $H-U_{n}h$ near $\{t=0,y=0\}$ to apply maximum principle.

By Theorem 7.3 Harnack inequality in slit domain, we have that $\frac{H}{U}\in C^{\alpha}(B_{1/2})$. In other words, $\exists a\in\mathbb{R}$ such that $|H-aU|\leq Cr^{\alpha}U$ in $B_{1/2}\setminus\{t\leq 0,y=0\}$. Therefore,

[TABLE]

By maximum principle, we have $-\epsilon U_{t}\leq H-U_{t}h\leq\epsilon U_{t}$ for any $\epsilon>0$ small. Letting $\epsilon\to 0$, we have the desired result.

Substep 2: Polynomial appromating $h$ in the whole space.

We apply the result of substep 1 to each slice $\{x^{\prime}=c\}$. It gives

[TABLE]

It suffices to show that all $b_{ml}^{h}(x^{\prime})$ are $x^{\prime}$ smooth functions and apply Taylor expansion to them at $x^{\prime}=0$. To show this, we consider $h^{\tau}(X)=\frac{h(X+\tau)-h(X)}{|\tau|}$ for $\tau$ in the $x^{\prime}$ direction. $h^{\tau}$ is a solution to (2.6), and $b_{ml}^{h^{\tau}}(x^{\prime})=\frac{b_{ml}^{h}(x^{\prime}+\tau)-b_{ml}^{h}(x^{\prime})}{|\tau|}$ by definition. Pass $|\tau|\to 0$. According to Remark 6.4, $D^{\tau}h$ is uniformly bounded. Therefore we have $\partial_{\tau}b_{ml}^{h}=b_{ml}^{\partial_{\tau}h}$ being bounded by universal constants and $k$. This gives the desired result.

Step 2: Properties of $P_{k}$.

To begin with, we show that $P_{k}$ is bounded by constant by $C(n,\beta,k)$. We notice that it suffices to prove the bound for $b_{ml}$ above. It is a direct result of Remark 6.4.

Next we prove the $L_{\beta}P=0$ by induction. Use $P_{k}$ to denote the approximating polynomial of order $k$. For $k=1$, the equality holds naturally. Assume the equality holds for $k$, we prove it for $k+1$.

We set $v=u-U_{0}P_{k}$. It satisfies that $L_{\beta}v=0$. By the expansion proven above, $v=U_{0}(P_{k+1}-P_{k}+o|X|^{k+1})$. We consider

[TABLE]

and pass $\lambda$ to [math]. It converges to $U_{0}(P_{k+1}-P_{k})$ uniformly. Therefore we have

[TABLE]

Finally, we notice that the $\alpha$ in Step 1 can be replaced by $1$. All we need to do is to consider the $k+1$ case, the $k+1+\alpha$ regularity is stronger than $k+1$ regularity. ∎

7. Appendix

In this Appendix, we gather some known results about flat boundaries, Harnack inequalities, and regularity theories. To begin with, we recall the Harnack type inequalities for the operator $L_{\beta}$. To begin with, we recall Harnack inequality. It will be used to estimate the solution $u$.

Theorem 7.1**.**

(Harnack inequality, [CRS10, Theorem 2.4] or [DSSS14, Theorem 2.13])

Let $u\geq 0$ be a solution of

[TABLE]

Then,

[TABLE]

for some constant $C$ depending only on $n$ and $\beta$.

Next we state the boundary Harnack inequality. Similar to Harnack inequality, boundary Harnack inequality will be used to analyze the behavior of $u$ near $\{u=0\}$.

Theorem 7.2**.**

(Boundary Harnack inequlity, [CRS10, Theorem 2.4 (ii)])

Assume that $u\geq 0$, $v$ satisfy

[TABLE]

then $\|\frac{v}{u}\|_{C^{\eta}}\leq C$ for some universal $\eta\in(0,1)$ and $C$.

Before we move on, we point out that the inequalites above can apply to a more general operator. For operator in the form div$(\varphi\nabla\cdot)$, the inequalites apply if for any ball $B$ we have $\sup(\frac{1}{|B|}\int_{B}\varphi)(\frac{1}{|B|}\int_{B}\varphi^{-1})\leq c$. We refer to [FJK82, FKS82] for details. Finally, we introduce Boundary Harnack in slit domains. It is used to analyze the behavior of $u$ near the free booundary. The pioneering paper [DS16] first provided the inequality for $\beta=0$. The paper [JN17] proved the inequlity for $-1<\beta<1$. A more recent paper [KW24] contains some more general Harnack inequalities.

Theorem 7.3**.**

(Boundary Harnack in slit domain, [JN17, Theorem 1.3])

Suppose that $\Omega\subset\mathbb{R}^{n}$ satisfy $\|\partial\Omega\|_{C^{k,\alpha}}\leq 1$, $0\in\partial\Omega$, and the normal unit vector of $\partial\Omega$ at [math] is $e_{n}$, the unit vector in $x_{n}$ direction. Furthermore, $u$ and $U>0$ satisfy

[TABLE]

Then

[TABLE]

for some $C=C(\beta,n,k,\alpha)>0$.

Finally, we introduce a sequence of regularity theorems to finish this Appendix. To begin with, we state a regularity theorem for the operator $L_{\beta}$. It is used to analyze the behavior of the modified function near $\{y=0\}$.

Proposition 7.4**.**

([JN17, Proposition 2.3]) Let $u$ be a solution to

[TABLE]

Then we have

[TABLE]

Furthermore, if $\bar{f}:=|y|^{\beta}\partial_{y}f\in L^{\infty}(B_{\lambda})$, then

[TABLE]

For the interior part, we will need the following Theorem. It is used to analyze the difference between the approximating function and paraboloid with the solution.

Theorem 7.5**.**

([STV21a, Theorem 1.2]) Suppose $u\in H^{1}{(|y|^{\beta}B_{1})}$ and $f\in L^{\infty}(B_{1})$ satisfy

[TABLE]

in the weak sense. For any $t\in(0,1)$, we have:

[TABLE]

where $C$ depends only on $n,t,\beta$.

For $f\equiv 0$, $t$ can be chosen to be exactly $1$. The estimate was proven in [CSS07], [CS07] and [FKS82]. We state it to end this paper.

Theorem 7.6**.**

([CS16, Proposition 3.5]) Let $u\in H^{1}(B_{1},y^{\beta}\,dX)$ be a weak solution to

[TABLE]

Then we have the following results.

1. (1)

For each $B_{r}(x_{0})\subset B_{1}$,

[TABLE]

where $M$ depends only on $n$ and $s$. 2. (2)

We have

[TABLE]

where $C$ depends only on $n$ and $s$. 3. (3)

For each integer $k\geq 0$ and each $B_{r}(x_{0})\subset B_{1}$,

[TABLE]

where $C$ depends only on $n$, $k$, and $s$.