Abstract
In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type problems and one phase problems.
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1 Introduction
In this paper, we prove some new pointwise boundary regularity for Dirichlet problems:
and oblique derivative problems:
where \(\Omega \subset \mathbb {R}^n\) is a bounded domain, \(0\in \partial \Omega \) and \(\beta \) is some given vector valued function on \(\partial \Omega \cap B_1\). The pointwise regularity shows a clear and deep relation between the regularity of solutions and the regularity of prescribed data. It can be tracked at least to the work of Caffarelli [3] for the interior pointwise regularity of fully nonlinear elliptic equations. Various pointwise regularity have been developed by many researchers since then, such as boundary regularity [12, 20], regularity for equations with lower terms [11, 15, 19,20,21], regularity for parabolic equations [22,23,24] and regularity for the Monge-Ampère equation [18] etc.
In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. We show that if the derivatives of u vanish, u possesses higher regularity than the usual. This was first observed in [12] and we give a complete treatment for the Dirichlet problems (1.1) and the oblique derivative problems (1.2) here.
As applications of these pointwise regularity, we prove the higher regularity of free boundaries in obstacle-type problems and one phase problems without using the partial hodograph-Legendre transformation (see [17]), which is a standard method up to now. We clarify the idea briefly. Take the Dirichlet problem (1.1) for instance. It is well-known that if \(\partial \Omega \in C^{k,\alpha }\) (\(k\ge 1\)), \(u\in C^{k,\alpha }\). On the other hand, the regularity of u may lead to the regularity of \(\partial \Omega \) since \(\varphi _i=-u_i/u_n\) (\(1\le i< n\)), where \(\varphi \) is the representation function of \(\partial \Omega \). If a problem is an overdetermined problem, i.e., we have more conditions on u on the boundary, we may have higher regularity for u and then higher regularity for \(\partial \Omega \) and so forth. Eventually, u and \(\partial \Omega \) are infinite smooth.
Before stating our main results, we introduce some notations for pointwise regularity. The first is the pointwise characterization of a function in Hölder spaces, which is well-known now.
Definition 1.1
Let \(U\subset \mathbb {R}^n\) be a bounded set and \(f:U\rightarrow \mathbb {R}\) be a function. We say that f is \(C^{k,\alpha }\) (\(k\ge 0, 0<\alpha \le 1\)) at \(x_0\in U\) or \(f\in C^{k, \alpha }(x_0)\) if there exist constants \(K,r_0>0\) and a polynomial \(P\in \mathcal {P}_k\) (i.e., degree less than or equal to k) such that
Then define \(D^if(x_0)=D^iP(x_0)\) (\(1\le i\le k\)),
If \(f\in C^{k, \alpha }(x)\) for any \(x\in U\) with the same \(r_0\) and
we say that \(f\in C^{k,\alpha }(\bar{U})\).
In addition, we say that f is \(C^{-1,\alpha }\) at \(x_0\) or \(f\in C^{-1,\alpha }(x_0)\) if there exist constants \(K,r_0>0\) such that
Then define
If \(f\in C^{-1, \alpha }(x)\) for any \(x\in U\) with the same \(r_0\) and
we say that \(f\in C^{-1,\alpha }(\bar{U})\).
Remark 1.2
If U is a smooth domain (e.g. a Lipschitz domain), the definition of \(f\in C^{k,\alpha }(\bar{U})\) (\(k\ge 0\)) is equivalent to the classical definition.
Remark 1.3
In this paper, we apply Definition 1.1 only to two kinds of sets:
(i) U is a domain, i.e., \(U=\Omega \cap B_1\) in our paper. We use this set for the solution u and the righthand term f since they are defined in \(\Omega \cap B_1\).
(ii) U is the boundary of a domain, i.e., \(U=\partial \Omega \cap B_1\) in our paper. We use this set for the boundary term g since it is only defined on \(\partial \Omega \cap B_1\).
If we use Definition 1.1 on \(\partial \Omega \cap B_1\), the polynomial P in (1.3) is not unique. For example, if
and P satisfies (1.3) on \(\partial \Omega \cap B_1\), then \(P+Q\cdot x_n\) also satisfies (1.3) for any polynomial Q.
Remark 1.4
In fact, the non-uniqueness of P is related to the boundary regularity. For instance, assume that \(\partial \Omega \in C^{1,\alpha }(0)\) (see Definition 1.5) and \(g=x_n\) on \(\partial \Omega \). We can regard g as a \(C^{\infty }(0)\) function since
On the other hand, we can regard g as a \(C^{1,\alpha }(0)\) function. Indeed, since \(\partial \Omega \in C^{1,\alpha }(0)\),
The benefit of the second viewpoint is that \(Dg(0)=0\), which is used for the boundary pointwise regularity (see Theorem 1.9).
The next is a pointwise characterization of the smoothness of a domain’s boundary. This definition is similar to Definition 1.1. That is, both definitions use polynomials to describe the smoothness. It was first introduced in [12].
Definition 1.5
Let \(\Omega \) be a bounded domain, \(\Gamma \subset \partial \Omega \) be relatively open and \(x_0\in \Gamma \). We say that \(\Gamma \) is \(C^{k,\alpha }\) (\(k\ge 0, 0<\alpha \le 1\)) at \(x_0\) or \(\Gamma \in C^{k,\alpha }(x_0)\) if there exist constants \(K,r_0>0\), a coordinate system \(\{x_1,...,x_n \}\) (isometric to the original coordinate system) and a polynomial \(P\in \mathcal {P}_k\) with \(P(0)=0\) and \(DP(0)=0\) (if \(k\ge 1\)) such that \(x_0=0\) in this coordinate system,
and
Then, define
and
If \(\Gamma \in C^{k, \alpha }(x)\) for any \(x\in \Gamma \) with the same \(r_0\) and
we say that \({\bar{\Gamma }}\in C^{k,\alpha }\). If \({\bar{\Gamma }}'\in C^{k,\alpha }\) for any \(\Gamma '\subset \subset \Gamma \), we say that \(\Gamma \in C^{k,\alpha }\). If \(\Gamma \in C^{k,\alpha }\) for any \(k\ge 1\) and \(0<\alpha \le 1\), we say that \(\Gamma \in C^{\infty }\).
Remark 1.6
The (1.5) and (1.6) means that the difference between \(\partial \Omega \cap B_{r_0}\) and P is controlled by \(K|x'|^{k+\alpha }\). Hence, this is an analogue of the pointwise \(C^{k,\alpha }\) for a function. One benefit of Definition 1.5 is that \(\partial \Omega \cap B_{r_0}\) doesn’t need to be the graph of some function. It could be rather complicated.
Remark 1.7
Throughout this paper, if we say that \(f\in C^{k,\alpha }(x_0)\) (\(\Gamma \in C^{k,\alpha }(x_0)\)), we use \(P_f\) (\(P_{\Omega }\)) to denote the corresponding polynomial in Definition 1.1 (Definition 1.5).
Remark 1.8
We always assume that \(0\in \partial \Omega \) and study the pointwise regularity at 0 for (1.1) and (1.2). In addition, if we use Definitions 1.1 and 1.5 at 0, we always assume that \(r_0=1\), and (1.5) and (1.6) hold if we say \(\partial \Omega \cap B_1\in C^{k,\alpha }(0)\).
We also use the following notation to describe the oscillation of \(\partial \Omega \) near 0. For \(r>0\), define
Now, we state our main results.
Theorem 1.9
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that for some integers \(k,l\ge 1\), \(u\in C^{k,\alpha }(0)\), \(f\in C^{k+l-2,\alpha }(0)\), \(g\in C^{k+l,\alpha }(0)\) and \(\partial \Omega \cap B_1\in C^{l,\alpha }(0)\). Moreover, assume that
Then \(u\in C^{k+l,\alpha }(0)\). That is, there exists \(P\in \mathcal {P}_{k+l}\) such that
and
where C depends only on \(n,k,l,\alpha \) and \(\Vert \partial \Omega \cap B_1\Vert _{C^{l,\alpha }(0)}\).
Remark 1.10
For the notion of viscosity solutions and related theories, we refer to [4, 5].
Remark 1.11
In this theorem, we assume that \(g\in C^{k+l,\alpha }(0)\). It means that there exists a polynomial \(P_g\) such that
If we take the polynomial corresponding to u (denoted by \(P_u\)), we of course have
and
As we have explained before, the polynomial for g is not unique. We can’t conclude that \(P_g\equiv P_u\). Hence, besides assuming that \(g\in C^{k+l,\alpha }(0)\), we must assume \(|g(0)|=\cdots =|D^k g(0)|=0\) additionally since \(P_g\ne P_u\).
Remark 1.12
In fact, the expression of the polynomial P can be written explicitly:
where \(a_{\sigma }\) are constants.
Remark 1.13
For equations with coefficients and lower order terms:
we also have \(u\in C^{k+l,\alpha }(0)\) (\(k\ge 2\)) if
Remark 1.14
Theorem 1.9 can be extended to more general equations, including fully nonlinear uniformly elliptic equations in general forms. For simplicity and clarity, we only consider the Laplace operator in this paper.
Remark 1.15
Note that Theorem 1.9 can only be stated in the form of pointwise regularity since we can’t propose an assumption like
Indeed, if u is harmonic in \(\Omega \cap B_1\) and \(u=|Du|=0\) on \(\partial \Omega \cap B_1\), then \(u\equiv 0\). This indicates that the assumption (1.10) is ill-poseness usually.
Remark 1.16
The (1.8) can be regarded as a polynomial version of (1.7).
As a consequence, we have the following boundary pointwise regularity.
Theorem 1.17
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that for some \(k\ge 1\), \(f\in C^{k-2,\alpha }(0)\), \(g\in C^{k,\alpha }(0)\) and \(\partial \Omega \cap B_1\in C^{k,\alpha }(0)\).
Then \(u\in C^{k,\alpha }(0)\). That is, there exists \(P\in \mathcal {P}_k\) such that
where C depends only on \(n,k,\alpha \) and \(\Vert \partial \Omega \cap B_1\Vert _{C^{k,\alpha }(0)}\). Moreover, if \(k=1\),
if \(k\ge 2\),
Remark 1.18
Theorem 1.17 has been proved in [11] as a special case. Since the result of Theorem 1.17 is not well-known and the proof for the Laplace operator is rather simple than that for fully nonlinear equations, we list this result and give a detailed proof in this paper.
As an application of Theorem 1.9 to the higher regularity of free boundaries in obstacle-type problems, we have
Theorem 1.19
Let u be a viscosity solution of
Assume that \(\partial \Omega \cap B_1\in C^{1,\alpha }\) for some \(0<\alpha <1\). Then \(u\in C^{\infty } ({\bar{\Omega }}\cap B_{1})\) and \(\partial \Omega \cap B_1\in C^{\infty }\).
Remark 1.20
Although we only consider the Poisson equation in this paper, the method is applicable to the higher regularity of free boundaries for fully nonlinear elliptic equations, which have been well studied (see [8]).
Remark 1.21
For the obstacle-type problem (1.11), one can prove the higher regularity starting from that \(\partial \Omega \) is Lipschitz continuous with the aid of boundary Harnack inequality (see [2, 16, Chapter 6.2]). Recently, De Silva and Savin [6] gave an elegant proof of the boundary Harnack inequality for equations in non-divergence form. Hence, we can prove the higher regularity in a simple way from Lipschitz regularity even for fully nonlinear elliptic equations.
Remark 1.22
Usually, the higher regularity of free boundaries is proved in the following way. First, by a proper partial hodograph-Legendre transformation (see [9]), (1.11) is transformed to an elliptic equation on a flat boundary and the free boundary \(\partial \Omega \cap B_1\) is represented by some function relation. Note that even for the Laplace operator, the transformed equation is a fully nonlinear elliptic equation in general. Hence, in the second step, one need to apply the theory for fully nonlinear elliptic equations (see [1]) to obtain the higher regularity of solutions and free boundaries.
In 2015, De Silva and Savin [7] gave a direct and simple proof of higher regularity of solutions and free boundaries based on a higher order boundary Harnack inequality. However, this method is not applicable to the fully nonlinear elliptic equations.
Remark 1.23
The Theorem 1.19 is a kind of rigidity theorem. It states that if we impose an additional condition: \(|Du|=0\) on \(\partial \Omega \cap B_1\) (besides \(u=0\) on \(\partial \Omega \cap B_1\)), u and \(\partial \Omega \) must be smooth. We can’t have one pointwise version of Theorem 1.19. Indeed, if we impose the pointwise condition: \(|u(0)|=|Du(0)|=0\), we can obtain only \(u\in C^{2,\alpha }(0)\) (by Theorem 1.9).
With respect to the boundary pointwise regularity for oblique derivative problems, we have
Theorem 1.24
Let \(0<\alpha <1\) and u be a viscosity solution of
where \(\beta \) is a vector valued function and satisfies the obliqueness condition: for some positive constant \(a_0\),
Suppose that \(f\in C^{-1,\alpha }(0)\), \(g\in C^{\alpha }(0)\), \(\beta \in C^{\alpha }(0)\) and \([\partial \Omega \cap B_1]_{C^{0,1}(0)}\le \delta \), where \(\delta >0\) depending only on \(n,\alpha \) and \(a_0\).
Then \(u\in C^{1,\alpha }(0)\). That is, there exists \(P\in \mathcal {P}_1\) such that
and
where C depends only on \(n,\alpha \) and \(a_0\).
Remark 1.25
We refer the reader to [10, 14] for the notion of viscosity solutions for oblique derivative problems.
Remark 1.26
Without loss of generality, for oblique derivative problems, we always assume that \(\Vert \beta \Vert _{L^{\infty }(\partial \Omega \cap B_1)}\le 1\) and (1.12) holds with the fixed constant \(a_0\) throughout this paper.
Remark 1.27
Theorem 1.24 shows that the boundary \(C^{1,\alpha }\) regularity holds on Lipschitz domains with a small Lipschitz constant.
For higher regularity, we have
Theorem 1.28
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that for some integers \(k,l\ge 1\), \(u\in C^{k,\alpha }(0)\), \(f\in C^{k+l-2,\alpha }(0)\), \(g\in C^{k+l-1,\alpha }(0)\), \(\beta \in C^{l-1,\alpha }(0)\) and \(\partial \Omega \cap B_1\in C^{l,\alpha }(0)\). Moreover, assume that
Then \(u\in C^{k+l,\alpha }(0)\). That is, there exists \(P\in \mathcal {P}_{k+l}\) such that
and
where C depends only on \(n,k,l,\alpha ,a_0\), \(\Vert \beta \Vert _{C^{l-1,\alpha }(0)}\) and \(\Vert \partial \Omega \cap B_1\Vert _{C^{l,\alpha }(0)}\).
Similar to the Dirichlet problem, we have the following higher order boundary pointwise regularity.
Theorem 1.29
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that for some \(k\ge 2\), \(f\in C^{k-2,\alpha }(0)\), \(g\in C^{k-1,\alpha }(0)\), \(\beta \in C^{k-1,\alpha }(0)\) and \(\partial \Omega \cap B_1\in C^{k-1,\alpha }(0)\).
Then \(u\in C^{k,\alpha }(0)\). That is, there exists \(P\in \mathcal {P}_k\) such that
and
where C depends only on \(n,k,\alpha ,a_0\), \(\Vert \beta \Vert _{C^{k-1,\alpha }(0)}\) and \(\Vert \partial \Omega \cap B_1\Vert _{C^{k-1,\alpha }(0)}\).
Remark 1.30
Similar to the Dirichlet problems, one can prove corresponding pointwise boundary regularity for fully nonlinear elliptic equations.
As an application of the above boundary pointwise regularity to the regularity of free boundaries in one phase problems, we have
Theorem 1.31
Let \(0<\alpha <1\) and u be a viscosity solution of
Assume that \(\partial \Omega \cap B_1\in C^{1,\alpha }\). Then \(u\in C^{\infty } ({\bar{\Omega }}\cap B_{1})\) and \(\partial \Omega \cap B_1\in C^{\infty }\).
In Sect. 2, we prove the boundary pointwise regularity for Dirichlet problems and the higher regularity for free boundaries of obstacle-type problems. Section 3 is devoted to the oblique derivative problems and higher regularity for free boundaries in one phase problems. The results of this paper show the underlying relation between the regularity of solutions and the regularity of boundaries. The proofs demonstrate how overdetermined conditions lead to higher regularity. Notations used in this paper are listed below, most of which are standard.
Notation 1.32
-
(1)
\(\{e_i\}^{n}_{i=1}\): the standard basis of \(\mathbb {R}^n\), i.e., \(e_i=(0,...0,\underset{i^{th}}{1},0,...0)\).
-
(2)
\(x'=(x_1,x_2,...,x_{n-1})\) and \(x=(x_1,...,x_n)=(x',x_n)\).
-
(3)
\(|x|:=\left( \sum _{i=1}^{n} x_i^2\right) ^{1/2}\) for \(x\in \mathbb {R}^n\).
-
(4)
\(\mathbb {R}^n_+=\{x\in \mathbb {R}^n\big |x_n>0\}\).
-
(5)
\(B_r(x_0)=B(x_0,r)=\{x\in \mathbb {R}^{n}\big | |x-x_0|<r\}\), \(B_r=B_r(0)\), \(B_r^+(x_0)=B_r(x_0)\cap \mathbb {R}^n_+\) and \(B_r^+=B^+_r(0)\).
-
(6)
\(T_r(x_0)\ =\{(x',0)\in \mathbb {R}^{n}\big | |x'-x_0'|<r\}\) and \(T_r=T_r(0)\).
-
(7)
\(A^c\): the complement of A; \({\bar{A}} \): the closure of A, where \( A\subset \mathbb {R}^n\).
-
(8)
\(\textrm{diam}(A)\): the diameter of A and \(\textrm{dist}(A,B)\): the distance between A and B, where \( A,B\subset \mathbb {R}^n\).
-
(9)
\(\Omega _r=\Omega \cap B_r\) and \((\partial \Omega )_r=\partial \Omega \cap B_r\).
-
(10)
\(\varphi _i=D_i \varphi =\partial \varphi /\partial x _{i}\), \(\varphi _{ij}=D_{ij}\varphi =\partial ^{2}\varphi /\partial x_{i}\partial x_{j}\) and we also use similar notations for higher order derivatives.
-
(11)
\(D^0\varphi =\varphi \), \(D \varphi =(\varphi _1,...,\varphi _{n} )\) and \(D^2 \varphi =\left( \varphi _{ij}\right) _{n\times n}\) etc.
-
(12)
\(|D^k\varphi |= \left( \sum _{|\sigma |=k}|D^{\sigma }\varphi | ^2\right) ^{1/2}\) for \(k\ge 1\), where the standard multi-index notation is used.
-
(13)
\(\mathcal {P}_k (k\ge 0):\) the set of polynomials of degree less than or equal to k. That is, any \(P\in \mathcal {P}_k\) can be written as
$$\begin{aligned} P(x)=\sum _{|\sigma |\le k}\frac{a_{\sigma }}{\sigma !}x^{\sigma } \end{aligned}$$where \(a_{\sigma }\) are constants. Define
$$\begin{aligned} \Vert P\Vert = \sum _{|\sigma |\le k}|a_{\sigma }|. \end{aligned}$$ -
(14)
\({\mathcal {H}}{\mathcal {P}}_k (k\ge 0):\) the set of homogeneous polynomials of degree k. That is, any \(P\in \mathcal{H}\mathcal{P}_k\) can be written as
$$\begin{aligned} P(x)=\sum _{|\sigma |= k}\frac{a_{\sigma }}{\sigma !}x^{\sigma }. \end{aligned}$$ -
(15)
\(\mathbf {\Pi }_k:\) The projection from \(\mathcal {P}_l\) to \(\mathcal {P}_k\) for \(l\ge k\). That is, if \(P\in \mathcal {P}_l\) is written as
$$\begin{aligned} P(x)=\sum _{|\sigma |\le l}\frac{a_{\sigma }}{\sigma !}x^{\sigma }, \end{aligned}$$then
$$\begin{aligned} \mathbf {\Pi }_kP(x)=\sum _{|\sigma |\le k}\frac{a_{\sigma }}{\sigma !}x^{\sigma }. \end{aligned}$$
2 Dirichlet problem and application to the obstacle problem
In this section, we give the proofs of Theorems 1.9, 1.17 and 1.19. We start with the following result (see [12, Corollary 4.3]):
Lemma 2.1
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that \(u\in C^{1,\alpha }(0)\), \(f\in C^{\alpha }(0)\), \(g\in C^{2,\alpha }(0)\) and \((\partial \Omega )_1\in C^{1,\alpha }(0)\). Moreover, assume that
Then \(u\in C^{2,\alpha }(0)\). That is, there exists \(P\in \mathcal{H}\mathcal{P}_2\) such that
and
where C depends only on \(n,\alpha \) and \(\Vert (\partial \Omega )_1\Vert _{C^{1,\alpha }(0)}\).
Proof
We sketch the proof for completeness. Since \(f\in C^{\alpha }(0)\) and \(g\in C^{2,\alpha }(0)\), we have
and
Since \(g(0)=|Dg(0)|=0\), \(P_g\in \mathcal{H}\mathcal{P}_2\).
Define
Then
and v satisfies
where
By (2.2) and (2.3) and noting \((\partial \Omega )_1\in C^{1,\alpha }(0)\),
From Theorem 4.2 in [12] (see also (4.10) and (4.11) there), there exists \(\tilde{P}\in \mathcal{H}\mathcal{P}_2\) such that
and
where C depends only on \(n,\alpha \) and \(\Vert (\partial \Omega )_1\Vert _{C^{1,\alpha }(0)}\).
Set
Then by the relation between u and v,
and
Therefore, the proof is completed. \(\square \)
Next, we prove a generalized version of Lemma 2.1:
Lemma 2.2
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that \(u\in C^{k,\alpha }(0) (k\ge 1)\), \(f\in C^{k-1,\alpha }(0)\), \(g\in C^{k+1,\alpha }(0)\) and \((\partial \Omega )_1\in C^{1,\alpha }(0)\). Moreover, assume that
Then \(u\in C^{k+1,\alpha }(0)\). That is, there exists \(P\in \mathcal{H}\mathcal{P}_{k+1}\) such that
and
where C depends only on \(n,k,\alpha \) and \(\Vert (\partial \Omega )_1\Vert _{C^{1,\alpha }(0)}\).
In the following proof, we will use a kind of homogeneous polynomial in a special form. We call \(Q\in \mathcal{H}\mathcal{P}_k\) a k-form (\(k\ge 1\)) if Q can be written as
That is, \(x_n\) appears in the expression of Q at least one time (thus \(Q\equiv 0\) on \(T_1\)), which turns out to be vital for the boundary regularity. In fact, \(P(x',0)\equiv P_g(x',0)\) in (2.4) indicates that \(P(x)-P_g(x)\) is a \((k+1)\)-form.
We prove Lemma 2.2 by induction. For \(k=1\), the lemma reduces to Lemma 2.1. Suppose that the lemma holds for \(k\le k_0-1\) and we need to prove the lemma for \(k=k_0\). First, we prove a key step towards the conclusion of Lemma 2.2.
Lemma 2.3
Let \(1\le k\le k_0\), \(0<\alpha <1\) and \(u\in C^{k,\alpha }(0)\) be a viscosity solution of
where \(P\in \mathcal{H}\mathcal{P}_{k-1}\). Suppose that
where \(\delta >0\) depends only on n, k and \(\alpha \).
Then there exists a \((k+1)\)-form Q such that
where \(C_0\) depends only on n and k, and \(\eta \) depends also on \(\alpha \).
Proof
We prove the lemma by contradiction. Suppose that the conclusion is false. Then there exist \(0<\alpha <1\) and sequences of \(u_m,f_m,g_m,\Omega _m,P_m\) (\(m\ge 1\)) satisfying \(u_m\in C^{k,\alpha }(0)\) and
In addition,
But for any \((k+1)\)-form Q satisfying \(\Vert Q\Vert \le C_0\) and \(\Delta Q+P_m\equiv 0\), we have
where \(C_0\) is to be specified later and \(0<\eta <1\) is taken small such that
Clearly, \(u_m\) are uniformly bounded (\(\Vert u_m\Vert _{L^{\infty }(\Omega _m\cap B_1)}\le 1\)). Moreover, \(u_m\) are equicontinuous (see [12, Lemma 2.7]). Hence, there exist \({\tilde{u}}:B_1^+\cup T_1\rightarrow \mathbb {R}\) and \({\tilde{P}}\in \mathcal{H}\mathcal{P}_{k-1}\) such that \(u_m\rightarrow {\tilde{u}}\) uniformly in compact subsets of \( B_1^+\cup T_1\), \(P_m\rightarrow {\tilde{P}}\) and
By the boundary \(C^{k,\alpha }\) estimate for \(u_m\) (Lemma 2.2 for \(k-1\) since \(k\le k_0\)) and noting \(u_m(0)=\cdots =|D^{k}u_m(0)|=0\), we have
Since \(u_m\) converges to u uniformly,
Hence, \({\tilde{u}}(0)=\cdots =|D^{k}{\tilde{u}}(0)|=0\). By the boundary estimate for \({\tilde{u}}\) on a flat boundary, there exists a \((k+1)\)-form \(\tilde{Q}\) such that
where \(C_0\) depends only on n and k.
Since \(P_m\rightarrow \tilde{P}\), there exist \((k+1)\)-forms \(\tilde{Q}_m\) such that \(\Delta (\tilde{Q}+ \tilde{Q}_m)+P_m\equiv 0\) and \(\Vert \tilde{Q}_m\Vert \rightarrow 0\) as \(m\rightarrow \infty \). Thus, (2.5) holds for \(Q=\tilde{Q}+\tilde{Q}_m\). Let \(m\rightarrow \infty \) in (2.5) and we have
which is a contradiction.
Now, we give the
Proof of Lemma 2.2
Since we have assumed that Lemma 2.2 holds for \(k_0-1\), \(u\in C^{k_0,\alpha }(0)\). By induction, we only need to prove Lemma 2.2 for \(k_0\), i.e., \(u\in C^{k_0+1,\alpha }(0)\). Without loss of generality, by a proper transformation, we assume that
for some \(P\in \mathcal{H}\mathcal{P}_{k_0-1}\) and
where \(\delta \) is as in Lemma 2.3 (with \(k=k_0\)) and \(C_1\) depending only on \(n,k_0\) and \(\alpha \) is to be chosen later.
Indeed, let \(u_1=u/K\) where
Then \(u_1\) satisfies
where \(f_1=f/K\) and \(g_1=g/K\).
Next, let \(u_2=u_1-P_{g_1}\). Since \(g(0)=|Dg(0)|\cdots =|D^{k_0}g(0)|=0\), \(P_g\in \mathcal{H}\mathcal{P}_{k_0+1}\). In addition, by \(u(0)=|Du(0)|\cdots =|D^{k_0}u(0)|=0\), we have \(f(0)=\cdots =|D^{k_0-2}f(0)|=0\) (see (2.4)). Hence, \(P_f\in \mathcal{H}\mathcal{P}_{k_0-1}\). Then \(u_2\) satisfies
where
In addition, \(P=\Delta P_{g_1}-P_{f_1}\in \mathcal{H}\mathcal{P}_{k_0-1}\) and
where C depends only on n and \(k_0\).
Finally, let \(y=x/\rho \) for \(\rho >0\) and \(\tilde{u}(y)=u_2(x)\). Then \(\tilde{u}\) satisfies
where
Hence,
Therefore, by taking \(\rho \) small enough (depending only on \(n,k_0,\alpha \) and \(\Vert (\partial \Omega )_1\Vert _{C^{1,\alpha }(0)}\)), the assumptions (2.8) and (2.9) for \(\tilde{u}\) can be guaranteed. Then the regularity of u can be derived from that of \(\tilde{u}\). Hence, without loss of generality, we assume that (2.8) and (2.9) hold for u.
To prove Lemma 2.2 for \(k_0\), we only need to show that there exist a sequence of \((k_0+1)\)-forms \(Q_m\) (\(m\ge 0\)) such that for all \(m\ge 1\),
and
where \(C_0\) and \(\eta \) are the constants as in Lemma 2.3.
We prove the above by induction. For \(m=1\), by Lemma 2.3 and setting \(Q_0\equiv 0\), the conclusion holds clearly. Suppose that the conclusion holds for m. We need to prove that the conclusion holds for \(m+1\).
Let \(r=\eta ^{m}\), \(y=x/r\) and
Then \(\tilde{u}\) satisfies
where
From (2.12), there exists \(C_1\) depends only on \(n,k_0\) and \(\alpha \) such that \(\Vert Q_i\Vert \le C_1\) (\(\forall ~0\le i\le m\)). By combining that \(Q_m\) is a \((k_0+1)\)-form and the definition of \(\partial \Omega \cap B_1\in C^{1,\alpha }(0)\) (see (1.5) and (1.6)), we have
Therefore,
By virtue of Lemma 2.3, there exists a \((k_0+1)\)-form \(\tilde{Q}\) such that
Let \(Q_{m+1}(x)=Q_m(x)+r^{k_0+1+\alpha } \tilde{Q}(y)=Q_m(x)+r^{\alpha }\tilde{Q}(x)\). Then (2.11) and (2.12) hold for \(m+1\). Recalling (2.13), we have
Hence, (2.10) holds for \(m+1\). By induction, the proof is completed. \(\square \)
Remark 2.4
By checking the proof, the condition on \(\partial \Omega \) is exactly used for estimating \(Q_m\) on \(\partial \Omega \) (see (2.14)). Hence, if the derivatives of u vanish, \(Q_m\) will be a higher order homogenous polynomial. This leads to a lower regularity assumption on \(\partial \Omega \). This is why we can obtain the \(C^{k_0+1,\alpha }\) regularity based only on \(\partial \Omega \in C^{1,\alpha }\).
Now, we can prove Theorem 1.9 based on Lemma 2.2.
Proof of Theorem 1.9
Throughout this proof, C always denotes a constant depending only on \(n,k,l,\alpha \) and \(\Vert (\partial \Omega )_1\Vert _{C^{l,\alpha }(0)}\). Without loss of generality, we assume
Since \(g\in C^{k+l,\alpha }(0)\),
Set \(u_1=u-P_g\) and then \(u_1\) is a viscosity solution of
where \(f_1=f-\Delta P_g\) and \(g_1=g-P_g\). Hence,
By Lemma 2.2, \(u_1\in C^{k+1,\alpha }(0)\) and there exists a \((k+1)\)-form \(Q_{k+1}\) such that
Let
where \(P_{\Omega }\in \mathcal {P}_l\) corresponds to \(\partial \Omega \) at 0 (note that \((\partial \Omega )_1\in C^{l,\alpha }(0)\)). Since \(Q_{k+1}\) is a \((k+1)\) -form and \(P_{\Omega }(0)=|DP_{\Omega }(0)|=0\),
Hence, \(u_2(0)=\cdots = |D^{k+1}u_2(0)|=0\). In addition, \(u_2\) satisfies
where \(f_2\in C^{k+l-2,\alpha }(0)\) and (note that \(Q_{k+1}\) is a \((k+1)\)-form and \((\partial \Omega )_1\in C^{l,\alpha }(0)\))
By Lemma 2.2 again, \(u_2\in C^{k+2,\alpha }(0)\) and there exists a \((k+2)\)-form \(Q_{k+2}\) such that
Let
Then \(u_3(0)=\cdots =|D^{k+2}u_3(0)|=0\). In addition, \(u_3\) is a viscosity solution of
where \(f_3\in C^{k+l-2,\alpha }(0)\) and
where.
By Lemma 2.2 again, \(u_3\in C^{k+2,\alpha }(0)\) and hence \(u\in C^{k+2,\alpha }(0)\). By similar arguments again and again, \(u\in C^{k+l,\alpha }(0)\) eventually and (1.8) holds. Therefore, the proof of Theorem 1.9 is completed. \(\square \)
Remark 2.5
By checking the proof, we know that the polynomial P can be written as (1.9) (see (2.16)).
Next, we give the
Proof of Theorem 1.17
For \(k=1\), Theorem 1.17 reduces to Lemma 2.1. For \(k\ge 2\), \(u\in C^{1,\alpha }(0)\) of course. Let
Note that
Hence, \(\tilde{u}(0)=|D\tilde{u}(0)|=0\). In addition, \(\tilde{u}\) satisfies
where \(\tilde{f}\in C^{k-2,\alpha }(0)\), \(\tilde{g}\in C^{k,\alpha }(0)\) and
Thus, \(\tilde{g}(0)=|D\tilde{g}(0)|=0\). By Theorem 1.9, \(\tilde{u}\) and hence \(u\in C^{k,\alpha }(0)\). \(\square \)
Finally, we prove the higher regularity of free boundaries with the aid of the boundary pointwise regularity.
Proof of Theorem 1.19
Assume that
where \(\varphi \in C^{1,\alpha }(T_1)\) and \(\varphi (0)=|D\varphi (0)|=0\). Since \(u=|Du|=0\) on \((\partial \Omega )_1\) and \((\partial \Omega )_1\in C^{1,\alpha }\), by Theorem 1.9, \(u\in C^{2,\alpha }(x_0)\) for any \(x_0\in (\partial \Omega )_1\). By combining with the interior regularity, \(u\in C^{2,\alpha }({\bar{\Omega }}')\) for any \(\Omega '\subset \subset {\bar{\Omega }}\cap B_1\).
From \(|D\varphi (0)|=0\) and \(u=|Du|=0\) on \((\partial \Omega )_1\) again, \(u_{ij}(0)=0\) for \(i+j<2n\). Hence, \(u_{nn}(0)=1\) by the equation \(\Delta u=1\). Let \(v(x)=u(x)-x_n^2/2\) and then v satisfies
where
That is, \(g\in C^{2,2\alpha }(0)\) and \(g(0)=|Dg(0)|=|D^2\,g(0)|=0\). By Theorem 1.9, \(v\in C^{2,2\alpha }(0)\) and hence \(u\in C^{2,2\alpha }(0)\). Similarly, for any \(x_0\in \partial \Omega \cap B_1\), \(u\in C^{2,2\alpha }(x_0)\). Hence, \(u\in C^{2,2\alpha } ({\bar{\Omega }}')\) for any \(\Omega '\subset \subset {\bar{\Omega }}\cap B_1\).
Since \(u_{nn}(0)=1\), \(u_{nn}\ge 1/2\) in \(\Omega _r\) for some \(r>0\). Then \(u_i/u_n\in C^{2\alpha } ({\bar{\Omega }}\cap B_{r})\). Note that \(\varphi _i=-u_i/u_n\). Thus, \(\varphi \in C^{1,2\alpha }(T_r)\), i.e., \((\partial \Omega )_r\in C^{1,2\alpha }\). By considering other \(x_0\in (\partial \Omega )_1\) similarly, we have \((\partial \Omega )_1\in C^{1,2\alpha }\).
Consider v again and \(g\in C^{2,4\alpha }(0)\) now (since \((\partial \Omega )_1\in C^{1,2\alpha }\)). From Theorem 1.9, \(v\in C^{2,4\alpha }(0)\). By similar arguments as above, \(u\in C^{2,4\alpha }({\bar{\Omega }}')\) for any \(\Omega '\subset \subset \bar{\Omega }\cap B_1\). Therefore, \((\partial \Omega )_1\in C^{1, 4\alpha }\). Consider v again and again and we have \((\partial \Omega )_1\in C^{2,\tilde{\alpha }}\) for some \(0<\tilde{\alpha }<1\) eventually.
Let \(v(x)=u(x)-(x_n-P_{\Omega }(x'))^2/2\) where \(P_{\Omega }\in \mathcal {P}_2\) is the polynomial corresponding to \(\partial \Omega \) at 0 since \((\partial \Omega )_1\in C^{2,\tilde{\alpha }}\). Then v satisfies
where \(f\in \mathcal {P}_2\) and
As before, by Theorem 1.9, \(v\in C^{4, \tilde{\alpha }}(0)\) and hence \(u\in C^{4, \tilde{\alpha }}(0)\). Similar to previous arguments, \(u\in C^{4,\tilde{\alpha }}({\bar{\Omega }}')\) for any \(\Omega '\subset \subset \bar{\Omega } \cap B_1\) and then \((\partial \Omega )_1\in C^{3,\tilde{\alpha }}\).
Let \(v(x)=u(x)-(x_n-P_{\Omega }(x'))^2/2\) where \(P_{\Omega }\in \mathcal {P}_3\) now. Repeat above arguments and we have \(u\in C^{\infty }({\bar{\Omega }}\cap B_1)\) and \((\partial \Omega )_1 \in C^{\infty }\) eventually. \(\square \)
Remark 2.6
In fact, we need a variation of Theorem 1.9 in the proof. That is, if \(g\in C^{k+l,\alpha }(0)\) is replaced by \(g\in C^{k+\tilde{l}, {\tilde{\alpha }}}(0)\) with \(\tilde{l}+\tilde{\alpha }\le l+\alpha \), we have \(u\in C^{k+\tilde{l},{\tilde{\alpha }}}(0)\). This variation can be proved by almost the same proof. For clarity, we only give Theorem 1.9 with \(\tilde{l}=l\) and \(\tilde{\alpha }=\alpha \).
Remark 2.7
Maybe a more natural idea of proving Theorem 1.19 is to consider Du instead of \(u-x_n^2/2\). The \(u\in C^{2,\alpha }\) is easy to obtain. If \(Du\in C^{2,\alpha }\) as well, \(u\in C^{3,\alpha }\) and then \((\partial \Omega )_1\in C^{2,\alpha }\). By a series of iteration, the proof is completed.
For \(1\le i\le n-1\), \(u_i=0\) on \((\partial \Omega )_1\) and \(Du_i(0)=0\). Hence, \(u_i\in C^{2,\alpha }(0)\) by Theorem 1.9. However, we can’t obtain \(u_n\in C^{2,\alpha }(0)\) since \(Du_n(0)=e_n\ne 0\). In addition, we can’t obtain \(u_i\in C^{2,\alpha }(x_0)\) for any \(x_0\in (\partial \Omega )_1\) since \(Du_i(x_0)\ne 0\) for other \(x_0\).
3 Oblique derivative problem and application to the one-phase problem
In this section, we give the detailed proofs of Theorems 1.24–1.31. As in the proofs for Dirichlet problems, we intend to use compactness method to prove the regularity of solutions. Hence, we need to build a uniform estimate for solutions. First, we prove a Harnack type inequality.
Lemma 3.1
Let \(u\ge 0\) be a viscosity solution of
Suppose that \(\underset{B_1}{\textrm{osc}}~\partial \Omega \le \delta \le \rho /8\), where \(0<\rho <1\) depends only on n and \(a_0\).
Then for any \(4\delta /\rho \le R\le 1/2\), we have
where C depends only on n and \(a_0\),
and
Proof
By the interior Harnack inequality,
where C depends only on n and \(\rho \). Hence, to prove (3.1), we only need to show
Without loss of generality, we assume that \(\inf _{\tilde{G}(R)}u=1\).
Let
Then it can be verified easily that (by taking \(\rho \) small enough)
Indeed, only the last inequality require some calculation. Since \(\beta _n\ge a_0\), \(\Vert \beta \Vert _{L^{\infty }}\le 1\), \(R\ge 4\delta /\rho \) and \(\underset{B_1}{\textrm{osc}}~\partial \Omega \le \delta \), we have
provided \(\rho \le a_0/(16n)\).
Let \(w=u-v\) and then
By the Alexandrov-Bakel’man-Pucci maximum principle for oblique derivative problems (see [10, Theorem 2.1]),
where C depends only on n and \(a_0\). Hence, by noting
we have
That is, (3.2) holds. \(\square \)
Remark 3.2
For the Dirichlet problem, we can obtain the equicontinuity up to the boundary by constructing proper barriers. In contrast, for the oblique derivative problem, we use the Harnack type inequality to show the equicontinuity since the solutions satisfies an equation on the boundary, which indicates that we should adopt the technique for the interior regularity rather than the technique for the boundary regularity (e.g. constructing barrier functions).
Remark 3.3
Note that (3.1) is not a true Harnack inequality since it holds for \(4\delta /\rho \le R\le 1/2\) other than \(0<R\le 1/2\). However, it is sufficient to provide the compactness in the proof (see Lemma 3.7 below) and requires less smoothness of \(\partial \Omega \).
Remark 3.4
The construction of the auxiliary function v is motivated by [13] (see Lemma 2.1 there) and has been used in [10] (see Theorem 2.2 there).
By a standard iteration argument, Lemma 3.1 implies the following uniform estimate.
Corollary 3.5
Let u be a viscosity solution of
Suppose that \(\Vert u\Vert _{L^{\infty }(\Omega _1)}\le 1\), \(\Vert f\Vert _{L^{n}(\Omega _1)}\le 1\), \(\Vert g\Vert _{L^{\infty }((\partial \Omega )_1)}\le 1\) and \(\underset{B_1}{\textrm{osc}}~\partial \Omega \le \delta \le \rho /8\), where \(\rho \) is as in Lemma 3.1.
Then for \(4\delta /\rho \le R\le 1/2\),
where \(0<\alpha <1\) is a universal constant and C depends only on n and \(a_0\).
Based on the above uniform estimate, we have the following equicontinuity for solutions.
Lemma 3.6
For any \(\Omega '\subset \subset \bar{\Omega }\cap B_1\) and \(\varepsilon >0\), there exists \(\delta >0\) (depending only on \(n,a_0,\varepsilon \) and \(\Omega '\)) such that if u is a viscosity solution of
with \(\Vert u\Vert _{L^{\infty }(\Omega _1)}\le 1\), \(\Vert f\Vert _{L^{n}(\Omega _1)}\le 1\), \(\Vert g\Vert _{L^{\infty }((\partial \Omega )_1)}\le 1\) and \(\underset{B_1}{\textrm{osc}}~\partial \Omega \le \delta \), then for any \(x,y\in \Omega '\) with \(|x-y|\le \delta \),
Proof
For any \(\Omega '\subset \subset \bar{\Omega }\cap B_1\), \(\varepsilon >0\) and \(x,y\in \Omega '\), let \(\delta >0\) to be specified later. Take \(x_0\in (\partial \Omega )_1\) such that \(|x-x_0|=\textrm{dist}(x,(\partial \Omega )_1)\). By Corollary 3.5 (a scaling version in fact), there exists \(\delta _1>0\) (small enough) depending only on \(n,a_0,\varepsilon \) and \(\Omega '\) such that if \(\underset{B_1}{\textrm{osc}}~\partial \Omega \le \rho \delta _1/4\), \(|x-x_0|\le \delta _1\) and \(|y-x_0|\le \delta _1\), we have
If \(|x-x_0|> \delta _1\), by the interior Lipschitz estimate for harmonic functions,
where C depends only on n.
Take \(\delta \) small enough such that \(\delta \le \rho \delta _1/4\) and \(C\delta /\delta _1\le \varepsilon /2\). Then by combining (3.5) and (3.6), we derive the conclusion. \(\square \)
In the following, we prove the boundary pointwise regularity for oblique derivative problems. First, we prove a key step.
Lemma 3.7
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that \(\Vert u\Vert _{L^{\infty }(\Omega _1)}\le 1\), \(\Vert f\Vert _{L^{n}(\Omega _1)}\le \delta \), \(\Vert g\Vert _{L^{\infty }((\partial \Omega )_1)} \le \delta \), \(\Vert \beta -\beta (0)\Vert _{L^{\infty }((\partial \Omega )_1)}\le \delta \) and \(\underset{B_1}{\textrm{osc}}~ \partial \Omega \le \delta \), where \(0<\delta <1\) depends only on \(n,a_0\) and \(\alpha \).
Then there exists \(P\in \mathcal {P}_1\) such that
where \(C_0\) depends only on n and \(a_0\), and \(\eta \) depends also on \(\alpha \).
Proof
We prove the lemma by contradiction. Suppose that the lemma is false. Then there exist \(0<\alpha <1\) and sequences of \(u_m,f_m,g_m,\beta _m,\Omega _m\) such that
with \(\Vert u_m\Vert _{L^{\infty }(\Omega _m\cap B_1)}\le 1\), \(\Vert f_m\Vert _{L^{n}(\Omega _m\cap B_1)}\le 1/m\), \(\Vert g_m\Vert _{L^{\infty }(\partial \Omega _m\cap B_1)}\le 1/m\), \(\Vert \beta _m-\beta _m(0)\Vert _{L^{\infty }((\partial \Omega )_1)}\le 1/m\) and \(\underset{B_1}{\textrm{osc}}~\partial \Omega _m\le 1/m\). In addition, for any \(P\in \mathcal {P}_1\) with \(\Vert P\Vert \le C_0\) and \(\beta _m(0)\cdot DP=0\),
where \(C_0\) is to be specified later and \(0<\eta <1\) is taken small such that
Note that \(u_m\) are uniformly bounded (\(\Vert u_m\Vert _{L^{\infty }(\Omega _m\cap B_1)}\le 1\)). Moreover, by Lemma 3.6, \(u_m\) are equicontinuous. Precisely, for any \(\Omega '\subset \subset B_1^+\cup T_1\), \(\varepsilon >0\), there exist \(\delta >0\) and \(m_0\) such that for any \(m\ge m_0\) and \(x,y\in \Omega '\cap \bar{\Omega }_m\) with \(|x-y|<\delta \), \(|u_m(x)-u_m(y)|\le \varepsilon \). Hence, there exists a subsequence (denoted by \(u_m\) again) such that \(u_m\) converges uniformly to some continuous function \(\tilde{u}\) on compact subsets of \(B_1^+\cup T_1\). Furthermore, there exists \(\beta ^0\) with \(\beta ^0_n\ge a_0\) such that \(\beta _m(0)\rightarrow \beta ^0\). By the closedness of viscosity solutions (e.g., see [10, Proposition 2.1.]), \(\tilde{u}\) is a viscosity solution of
By the boundary regularity for homogeneous equations on flat boundaries (e.g. see [10, Theorem 4.1 and Theorem 4.2]), there exists \({\tilde{P}}\in \mathcal {P}_1\) such that
Since \(\beta _m(0)\rightarrow \beta ^0\), there exists \(P_m\in \mathcal{H}\mathcal{P}_1\) such that \(\beta _m(0)\cdot (D\tilde{P}+DP_m)=0\) and \(\Vert P_m\Vert \rightarrow 0\) as \(m\rightarrow \infty \). Thus, (3.7) holds for \(\tilde{P}+P_m\). Let \(m\rightarrow \infty \) in (3.7) and we have
On the other hand, from (3.8) and (3.9),
which is a contradiction. \(\square \)
Remark 3.8
Usually, to prove Lemma 3.7, we solve an equation to approximate u (see the proofs of Lemmas 5.1 and 6.3 in [10]). Instead, the compactness method avoids the solvability. This is one of the main advantages of the method of compactness.
Remark 3.9
Note that Lemma 3.6 is not a true equicontinuity up to the boundary. However, it is enough to provide the compactness in the proof of Lemma 3.7. The benefit is that we don’t require the smoothness of \(\partial \Omega \cap B_1\) and hence we can develop the pointwise regularity.
Now, we can prove the boundary pointwise \(C^{1,\alpha }\) regularity.
Theorem 3.10
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that \(f\in C^{-1,\alpha }(0)\), \(g\in C^{\alpha }(0)\), \(\beta \in C^{\alpha }(0)\) and \([\partial \Omega \cap B_1]_{C^{0,1}(0)} \le \delta \), where \(\delta \) is as in Lemma 3.7.
Then u is \(C^{1,\alpha }\) at 0, i.e., there exists \(P\in \mathcal {P}_1\) such that
and
where C depends only on \(n,a_0,\alpha \) and \(\Vert \beta \Vert _{C^{\alpha }(0)}\).
Proof
We assume that \(g(0)=0\). Otherwise, we may consider \(\tilde{u}=u-g(0)x_n/\beta _n(0)\). Let \(\delta \) be as in Lemma 3.7, which depends only on \(n,a_0\) and \(\alpha \). Without loss of generality, we assume that
where \(C_1\) is a constant (depending only on \(n,a_0\) and \(\alpha \)) to be specified later.
To show \(u\in C^{1,\alpha }(0)\), we only need to prove that there exists a sequence of \(P_m\in \mathcal {P}_1\) (\(m\ge -1\)) such that for all \(m\ge 0\),
and
where \(C_0\) and \(\eta \) are constants as in Lemma 3.7.
We prove the above by induction. For \(m=0\), by setting \(P_0\equiv P_{-1}\equiv 0\), the conclusion holds clearly. Suppose that the conclusion holds for m. We need to prove that the conclusion holds for \(m+1\).
Let \(r=\eta ^{m}\), \(y=x/r\) and
Then \(\tilde{u}\) satisfies
where
By (3.14), there exists a constant \(C_1\) depending only on \(n,a_0\) and \(\alpha \) such that \(|DP_i|\le C_1\) (\(\forall ~0\le i\le m\)). Then it is easy to verify that
From 3.7, there exists \(\tilde{P}\in \mathcal {P}_1\) such that
Let \(P_{m+1}(x)=P_{m}(x)+r^{1+\alpha }\tilde{P}(y)\). Then (3.14) and (3.15) hold for \(m+1\). Recalling (3.16), we have
Hence, (3.13) holds for \(m+1\). By induction, the proof is completed. \(\square \)
Next, we prove a lemma similar to Lemma 2.2.
Lemma 3.11
Let \(0<\alpha <1\) and u be a viscosity solution of
Suppose that \(u\in C^{k,\alpha }(0)(k\ge 1)\), \(f\in C^{k-1,\alpha }(0)\), \(g\in C^{k,\alpha }(0)\), \(\beta \in C^{\alpha }(0)\) and \((\partial \Omega )_1\in C^{1,\alpha }(0)\). Moreover, assume that
Then \(u\in C^{k+1,\alpha }(0)\). That is, there exists \(P\in \mathcal{H}\mathcal{P}_{k+1}\) such that
and
where C depends only on \(n,k,a_0,\alpha \), \(\Vert \beta \Vert _{C^{\alpha }(0)}\) and \(\Vert (\partial \Omega )_1\Vert _{C^{1,\alpha }(0)}\).
As in Sect. 2, we prove above lemma by induction. For \(k=1\), the lemma reduces to Theorem 3.10. Suppose that the lemma holds for \(k\le k_0-1\) and we need to prove the lemma for \(k=k_0\). First, we prove the following lemma which is a key step towards the conclusion of Lemma 3.11.
Lemma 3.12
Let \(1\le k\le k_0\), \(0<\alpha <1\) and \(u\in C^{k,\alpha }(0)\) be a viscosity solution of
where \(P\in \mathcal{H}\mathcal{P}_{k-1}\). Suppose that
where \(\delta >0\) depending only on \(n,k,a_0\) and \(\alpha \).
Then there exists \(Q\in \mathcal{H}\mathcal{P}_{k+1}\) such that
where \(C_0\) depends only on \(n,a_0\) and k, and \(\eta \) depends also on \(\alpha \).
Remark 3.13
Note that \(\beta (0)\cdot DQ(x',0)\equiv 0\) is equivalent to that \(\beta (0)\cdot DQ(x)\) is a k-form.
Proof
As before, we prove the lemma by contradiction. Suppose that the conclusion is false. Then there exist \(0<\alpha <1\) and sequences of \(u_m,f_m,g_m,\beta _m,\Omega _m, P_m\) (\(m\ge 1\)) satisfying \(u_m\in C^{k,\alpha }(0)\) and
In addition,
But for any \(Q\in \mathcal{H}\mathcal{P}_{k+1}\) satisfying
we have
where \(C_0\) is to be specified later and \(0<\eta <1\) is taken small such that
As in the proof of Lemma 3.7, \(u_m\) are uniformly bounded and equicontinuous. Hence, there exist \({\tilde{u}}:B_1^+\cup T_1\rightarrow \mathbb {R}\), \(\beta ^0\in \mathbb {R}^n\) with \(\beta ^0_n\ge a_0\) and \({\tilde{P}}\in \mathcal{H}\mathcal{P}_{k-1}\) such that \(u_m\rightarrow {\tilde{u}}\) uniformly in compact subsets of \( B_1^+\cup T_1\), \(\beta _m(0)\rightarrow \beta ^0\), \(P_m\rightarrow {\tilde{P}}\) and
By the boundary \(C^{k,\alpha }\) estimate for \(u_m\) (Lemma 3.11 for \(k-1\) since \(k\le k_0\)) and noting \(u_m(0)=\cdots =|D^{k}u_m(0)|=0\), we have
Since \(u_m\) converges to u uniformly,
Hence, \({\tilde{u}}(0)=\cdots =|D^{k}{\tilde{u}}(0)|=0\). By the boundary estimate for \({\tilde{u}}\), there exists \(\tilde{Q}\in \mathcal{H}\mathcal{P}_{k+1}\) such that
where \(C_0\) depends only on \(n,a_0\) and k.
Since \(\beta _m(0)\rightarrow \beta ^0\) and \(P_m\rightarrow \tilde{P}\), there exist \(Q_m\in \mathcal{H}\mathcal{P}_{k+1}\) such that \(\Vert Q_m\Vert \rightarrow 0\) and
Thus, (3.17) holds for \(Q=\tilde{Q}+Q_m\). Let \(m\rightarrow \infty \) in (3.17) and we have
However, by (3.18) and (3.19),
which is a contradiction. \(\square \)
Now, we give the
Proof of Lemma 3.11
Since we have assumed that Lemma 3.11 holds for \(k_0-1\), \(u\in C^{k_0,\alpha }(0)\). By induction, we only need to prove Lemma 3.11 for \(k_0\), i.e., \(u\in C^{k_0+1,\alpha }(0)\). Without loss of generality, by a proper transformation, we can assume as in the proof of Lemma 2.2 that
for some \(P\in \mathcal{H}\mathcal{P}_{k_0-1}\) and
where \(\delta \) is as in Lemma 3.12 (with \(k=k_0\)) and \(C_1\) depending only on \(n,k_0,a_0\) and \(\alpha \) is to be chosen later.
To prove Lemma 3.11 for \(k_0\), we only to show that there exists a sequence of \(Q_m\in \mathcal{H}\mathcal{P}_{k_0+1}\) (\(m\ge 0\)) such that for all \(m\ge 1\),
and
where \(C_0\) and \(\eta \) are the constants as in Lemma 3.12.
We prove the above by induction. For \(m=1\), by Lemma 3.12 and setting \(Q_0\equiv 0\), the conclusion holds clearly. Suppose that the conclusion holds for m. We need to prove that the conclusion holds for \(m+1\).
Let \(r=\eta ^{m}\), \(y=x/r\) and
Then \(\tilde{u}\) satisfies
where
By (3.22), there exists \(C_1\) depends only on \(n,k_0,a_0\) and \(\alpha \) such that \(\Vert Q_i\Vert \le C_1\) (\(\forall ~0\le i\le m\)). Since \(\beta (0)\cdot DQ_m\) is a \(k_0\)-form (see (3.23)) and \((\partial \Omega )_1\in C^{1,\alpha }(0)\),
Then
By Lemma 3.12, there exists \(\tilde{Q}\in \mathcal{H}\mathcal{P}_{k_0+1}\) such that
Let \(Q_{m+1}(x)=Q_m(x)+r^{k_0+1+\alpha }\tilde{Q}(y)=Q_m(x)+r^{\alpha }\tilde{Q}(x)\). Then (3.22) and (3.23) hold for \(m+1\). By recalling (3.24), we have
Hence, (3.21) holds for \(m+1\). By induction, the proof is completed. \(\square \)
Next, we prove Theorem 1.28 with the aid of Lemma 3.11.
Proof of Theorem 1.28
Throughout this proof, C always denotes a constant depending only on \(n,k,l,a_0,\alpha \), \(\Vert \beta \Vert _{C^{l-1,\alpha }(0)}\) and \(\Vert (\partial \Omega )_1\Vert _{C^{l,\alpha }(0)}\). Without loss of generality, we assume as before
Since \(g\in C^{k+l-1,\alpha }(0)\) and \(\beta \in C^{l-1,\alpha }(0)\),
and
Note that \(P_g\in \mathcal{H}\mathcal{P}_{k}\) and \(P_{\beta }\) is a vector valued polynomial. Take a polynomial \(P_0\in \mathcal{H}\mathcal{P}_{k+l}\) such that
Set \(u_1=u-P_0\) and then \(u_1\in C^{k,\alpha }(0)\) and \(u_1(0)=\cdots =|D^ku_1(0)|=0\). In addition, \(u_1\) is a viscosity solution of
where \(f_1=f-\Delta P_0\) and
Hence, by (3.26), (3.27) and noting \(P_0\in \mathcal{H}\mathcal{P}_{k+l}\),
By Lemma 3.11, \(u_1\in C^{k+1,\alpha }(0)\). That is, there exists \(P_{k+1}\in \mathcal{H}\mathcal{P}_{k+1}\) such that \(\beta (0)\cdot DP_{k+1}\) is a k-form and
Let
where \(P_{\Omega }\in \mathcal {P}_{l}\) corresponds to \(\partial \Omega \) at 0 (note that \((\partial \Omega )_1\in C^{l,\alpha }(0)\)). Note that \(P_{\Omega }(0)=|DP_{\Omega }(0)|=0\) and then \(u_2(0)=\cdots =|D^{k+1}u_2(0)|=0\). In addition, \(u_2\) satisfies
where \(f_2\in C^{k+l-2,\alpha }(0)\),
and
By (3.27) and noting \(P_{k+1}\in \mathcal{H}\mathcal{P}_{k+1}\),
Since \(\beta (0)\cdot DP_{k+1}(x)\) is a k-form and \((\partial \Omega )_1\in C^{l,\alpha }(0)\),
Next, since \(\hat{g}\) is a polynomial, it can be verified easily that
Then by (3.28)–(3.31), we have \(g_2\in C^{k+l-1,\alpha }(0)\) and
By Lemma 3.11 again, \(u_2\in C^{k+2,\alpha }(0)\). That is, there exists \(P_{k+2}\in \mathcal{H}\mathcal{P}_{k+2}\) such that
and
I.e., \((\beta (0)\cdot DP_{k+2}-\mathbf {\Pi }_{k+1}(g_2))(x)\) is a \((k+1)\)-form.
Set
Hence, \(u_3(0)=\cdots =|D^{k+2}u_3(0)|=0\). In addition, \(u_3\) is a viscosity solution of
where \(f_3\in C^{k+l-2,\alpha }(0)\),
and
Similar to the above argument,
and
In addition, by (3.32), for any \(x\in (\partial \Omega )_1\),
Hence,
That is, \(g_3(0)=\cdots =|D^{k+1}g_3(0)|=0.\) By virtue of Lemma 3.11 again, \(u_3\in C^{k+3,\alpha }(0)\). By similar arguments again and again, \(u\in C^{k+l,\alpha }(0)\) eventually and (1.13) holds. Therefore, the proof of Theorem 1.28 is completed. \(\square \)
The Theorem 1.29 is an easy consequence of Theorem 1.28.
Proof of Theorem 1.29
In fact, we can always assume that \(u(0)=|Du(0)|=0\) since the boundary condition is a first order equation. Let \(\tilde{u}(x)=u(x)-u(0)-Du(0)\cdot x\) and then \(\tilde{u}\) satisfies
where \(\tilde{g}=g-\beta \cdot Du(0)\). Note that \(\beta (0)\cdot Du(0)=g(0)\) and hence
Thus, \(\tilde{u}(0)=|D\tilde{u}(0)|=\tilde{g}(0)=0\) and \(\tilde{g}\in C^{k-1,\alpha }(0)\). By Theorem 1.28 with \(l=1\), we arrive at the conclusion of Theorem 1.29.
Finally, as an application to the regularity of free boundaries in one phase problems, we prove Theorem 1.31.
Proof of Theorem 1.31
Since \(u=0\) on \(\partial \Omega \cap B_1\) and \((\partial \Omega )_1\in C^{1,\alpha }\), by the boundary pointwise regularity for Dirichlet problems, \(u\in C^{1,\alpha }(\bar{\Omega }\cap B_1)\). Hence, we have \(|Du|=1\) on \((\partial \Omega )_1\) in the classical sense. Let \(v(x)=u(x)-x_n\) and then
Since \(u\in C^{1,\alpha }(0)\), \(u_i\in C^{\alpha }(0)\) and hence \(u_i^2\in C^{2\alpha }(0)\). By the boundary pointwise regularity for oblique derivative problems (see Theorem 1.24), \(v\in C^{1, 2\alpha }(0)\) and then \(u\in C^{1, 2\alpha }(0)\). Similarly, \(u\in C^{1, 2\alpha }(x_0)\) for any \(x_0\in (\partial \Omega )_1\). Note that \(u_n(0)=1\) and then \(u_i/u_n\in C^{2\alpha }({\bar{\Omega }} \cap B_r)\) for some \(r>0\). Thus, \(\partial \Omega \cap B_r\in C^{1,2\alpha }\). Likewise, \((\partial \Omega )_1\in C^{1,2\alpha }\).
Repeat above argument. Note \(u_i^2\in C^{4\alpha }(0)\). By Theorem 1.24, \(v\in C^{1, 4\alpha }(0)\) and we have \((\partial \Omega )_1\in C^{1,4\alpha }\). After finite steps, \(u\in C^{2,{\tilde{\alpha }}}\) and \((\partial \Omega )_1\in C^{2,{\tilde{\alpha }}}\) for some \(0<{\tilde{\alpha }}<1\).
Note that \(u_i(0)=0\) for \(1\le i\le n-1\). Then \(u_i^2\in C^{2,{\tilde{\alpha }}}(0)\). By Theorem 1.29, \(v\in C^{3,{\tilde{\alpha }}}(0)\). Hence \(u\in C^{3,{\tilde{\alpha }}}(\Omega _1)\) and \((\partial \Omega )_1\in C^{3,{\tilde{\alpha }}}\). Thus, \(u_i^2\in C^{3,{\tilde{\alpha }}}\) then \(u\in C^{4,{\tilde{\alpha }}}\) and \((\partial \Omega )_1\in C^{4,{\tilde{\alpha }}}\). By iteration arguments, we have \(u\in C^\infty \) and \((\partial \Omega )_1\in C^{\infty }\) eventually. \(\square \)
Remark 3.14
Since \(u=0\) and \(|Du|=1\) on \((\partial \Omega )_1\), \(\partial u/\partial \nu =1\) on \((\partial \Omega )_1\) where \(\nu \) is the inner normal. Thus, maybe a more natural idea of proving Theorem 1.31 is to consider the Neumann problem:
However, the smoothness of \(\nu \) depends on the smoothness of \((\partial \Omega )_1\). Hence, we can’t improve the regularity of solutions and \((\partial \Omega )_1\). Instead, \(\beta \equiv e_n\) if considering (3.33).
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Lian, Y., Zhang, K. Boundary pointwise regularity and applications to the regularity of free boundaries. Calc. Var. 62, 230 (2023). https://doi.org/10.1007/s00526-023-02573-9
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DOI: https://doi.org/10.1007/s00526-023-02573-9