1 Introduction

The Laplace equation arises widely in physics and engineering and is a classical prototype of partial differential equations. Consider the following Dirichlet problem for the Laplace equation:

$$ \textstyle\begin{cases} \Delta u=0 & \mbox{in } \Omega; \\ u=g & \mbox{on } \partial\Omega, \end{cases} $$
(1.1)

where \(\Omega\subset R^{n}\) is a bounded domain and g is a continuous function on ∂Ω.

In 1851, Riemann [1] proposed the famous Dirichlet principle, which states that there always exists a harmonic function continuous up to the boundary and coinciding with g on the boundary. However, Lebesgue [2] constructed a bounded domain on which the Dirichlet problem is not always solvable in 1912. By Perron’s method [3, 4], there always exists a harmonic function u in Ω with respect to g. If \(x_{0}\in\partial\Omega \) is a regular point (see [4, p.25] for the definition), then u is continuous up to \(x_{0}\). Hence, the solvability of (1.1) reduces to the problem whether the boundary points are regular. In 1924, Wiener [5] provided a sufficient and necessary condition for the regularity of \(x_{0}\). This is the famous Wiener criterion, which solves the Dirichlet problem completely.

However, the geometric interpretation of Wiener criterion is not clear, and it is not easy to verify whether a domain satisfies the Wiener criterion at some boundary point. One of the interesting cases is that the boundary of the domain near some boundary point is constructed by a spine. Precisely, suppose that for a proper coordinate system, \(0\in \partial\Omega\), and there exist \(0< r_{0}<1\) and a continuous nondecreasing function \(\varphi:R\rightarrow R\) with \(\varphi(0)=0\) such that

$$ B_{r_{0}}\cap\Omega= B_{r_{0}}\cap \bigl\{ x\in R^{n}| \bigl\vert x' \bigr\vert >\varphi(x_{n}) \mbox{ if } x_{n}>0 \bigr\} , $$
(1.2)

where \(B_{r}\) denotes the open ball in \(R^{n}\) with radius r and center 0, \(x=(x_{1},\ldots,x_{n})\in R^{n}\) and \(x'=(x_{1},\ldots,x_{n-1})\). If ‘=’ is replaced by ‘⊂’ in (1.2), we call that Ω satisfies the exterior spine condition with φ at 0.

It is natural to ask what condition φ should satisfy to guarantee that 0 is a regular point. It is well known that if \(\varphi (r)\geq Cr\) for some positive constant C, i.e., the domain satisfies the exterior cone condition, then 0 is a regular point (see [6] and [4, Problem 2.12]). In this note, we will consider the weaker condition and always assume that

$$ \varphi(r)/r\rightarrow0\quad \mbox{as } r\rightarrow0\quad \mbox{and}\quad \varphi (r)< r, \quad \forall 0< r< r_{0}. $$
(1.3)

In the dimension \(n=3\), if Ω satisfies (1.2), it is well known that 0 is a regular point if and only if φ satisfies (see Theorem 5.2 and the following example in [7])

$$ \int_{0}^{r_{0}} \frac{dr}{r|\ln\varphi(r)|} =+ \infty. $$
(1.4)

What condition φ should satisfy in higher dimensions is not known. Besides, the following boundary regularity results are well known. Suppose that g is identically zero on a portion of the boundary near 0. If \(\varphi(r)\geq Cr^{1/2}\), which implies that Ω satisfies the exterior sphere condition, then the solution is Lipschitz continuous at 0. If \(\varphi(r)\geq Cr\), i.e., Ω satisfies the exterior cone condition, then the solution is Hölder continuous at 0. A natural question is whether the condition \(\varphi (r) \geq Cr^{a}\) for some constant C and \(a>1\) can guarantee the continuity (even Hölder continuity) of the solution at 0. The geometric meaning of this condition is that Ω satisfies the exterior Hölder spine condition. It should be pointed out that for the dimension \(n=3\), the continuity of the solution in this case is guaranteed (see (1.4)).

This note is devoted to deriving the sufficient and necessary condition for φ in a higher dimension and answer the above question.

First, we recall the Wiener criterion. For any bounded domain \(\Omega \subset R^{n}\), its capacity is defined by (see Section 2.9 in [4] and Section 19 of Chapter XI in [7])

$$ \operatorname{cap} \Omega=\inf_{v\in K} \int|Dv|^{2}, $$

where

$$ K=\bigl\{ v\in C_{0}^{1}\bigl(R^{n}\bigr)|v=1 \mbox{ on } \Omega\bigr\} $$

and \(C_{0}^{1}(R^{n})\) denotes the set of functions having continuous derivatives and compact support in \(R^{n}\).

Suppose that \(0\in\partial\Omega\) and \(0<\lambda<r_{0}\) is a constant. Let

$$ \Omega_{j}= \bigl\{ x\notin\Omega| \lambda^{j+1}\leq|x|\leq \lambda^{j}\bigr\} \quad \mbox{and}\quad C_{j}= \operatorname{cap}\Omega_{j}\quad \mbox{for } j=1,2,\ldots. $$

The Wiener criterion states that 0 is a regular point if and only if the series

$$ \sum_{j=1}^{\infty} \frac{C_{j}}{\lambda^{j(n-2)}} $$
(1.5)

diverges (see Section 2.9 in [4], Section 19 of Chapter XI in [7] and Theorem 5.2 in [7]).

Our main result is the following theorem.

Theorem 1.1

Suppose that \(n\geq4\), \(0\in\partial\Omega\) and (1.2) is satisfied. Then 0 is a regular point with respect to (1.1) if and only if φ satisfies

$$ \int_{0}^{r_{0}} \frac{\varphi^{n-3}(r)\, dr}{r^{n-2}} =+ \infty. $$
(1.6)

An immediate consequence is the following.

Corollary 1.2

Let \(n\geq4\) and \(0\in\partial\Omega\). Suppose that Ω satisfies the exterior spine condition with φ at 0 and (1.6) holds.

Then 0 is a regular point with respect to (1.1).

Remark 1.3

From (1.6), the special dimensions \(n=2,3\) should be noted. In addition, for \(n\geq4\) and \(a>1\), \(\varphi(r) \geq Cr^{a}\) is not enough to guarantee that 0 is a regular point, which is an essential difference to dimensions 2 and 3.

2 Proof of Theorem 1.1

Now, we give the proof of Theorem 1.1.

Proof

We use ellipsoids to approximate \(\Omega_{j}\) and hence to estimate \(C_{j}\). Clearly, \(\Omega_{j}\) is contained in an ellipsoid \(E_{j}\) with semi-axes \(2\lambda^{j}\) and \(2\varphi(\lambda^{j})\) (\(n-1\) repeats), and \(\Omega_{j}\) contains an ellipsoid \(\tilde{E}_{j}\) with semi-axes \(\lambda^{j+1}\) and \(\varphi(\lambda^{j+1})/2\) (\(n-1\) repeats). The capacity for this kind ellipsoid E is (see (125) in [8])

$$ \operatorname{cap} E=\bigl(\beta^{2}- \gamma^{2}\bigr)^{(n-2)/2}\Big/ \int_{0}^{\arcsin(\sqrt {\mu})} \frac{\sin^{n-3}\theta}{\cos^{n-2}\theta}\, d\theta, $$
(2.1)

where β and γ are semi-axes of E with \(\beta>\gamma\) and \(\mu=1-\gamma^{2}/\beta^{2}\). Next, let

$$ I_{m,k}= \int_{0}^{\arcsin(\sqrt{\mu})} \frac{\sin^{m}\theta}{\cos ^{k}\theta}\, d\theta. $$

Then \(I_{m,k}\) has the following reduction formula for \(k\neq1\) (see (155) in [8]):

$$ I_{m,k}=\frac{\mu^{(m-1)/2}}{(k-1)(1-\mu)^{(k-1)/2}}-\frac{m-1}{k-1} I_{m-2,k-2}. $$

We say that \(A\simeq B\) if \(A_{1}A\leq B \leq A_{2} A\), where \(A_{1}\) and \(A_{2}\) are constants depending only on the dimension n. If \(1-\mu\ll 1\), then

$$ \frac{\mu^{(m-1)/2}}{(k-1)(1-\mu)^{(k-1)/2}}\simeq(1-\mu)^{-(k-1)/2}. $$

Substitute \(m=n-3\) and \(k=n-2\) and, by noting for \(k=1\) (see [9, Section 442.10])

$$ I_{0,1}=\frac{1}{2}\ln\frac{1+\sqrt{\mu}}{1-\sqrt{\mu}}, $$

we have

$$ I_{n-3,n-2}\simeq(1-\mu)^{-(n-3)/2}. $$

Therefore, substituting into (2.1) leads to

$$ \operatorname{cap} E\simeq\bigl(\beta^{2}-\gamma^{2} \bigr)^{(n-2)/2} \frac{\gamma ^{n-3}}{\beta^{n-3}}\simeq\beta\gamma^{n-3}. $$

Let \(E=E_{j}\) with \(\beta=2\lambda^{j}\) and \(\gamma=2\varphi(\lambda^{j})\), then \(1-\mu\ll1\) for sufficient large j (recall (1.3)) and hence

$$ C_{j}=\operatorname{cap} \Omega_{j}\leq\operatorname{cap} E_{j} \leq A_{1} \lambda^{j} \varphi^{n-3} \bigl(\lambda^{j}\bigr). $$

Similarly, let \(E=\tilde{E}_{j}\) with \(\beta=\lambda^{j+1}\) and \(\gamma =\varphi(\lambda^{j+1})/2\), then

$$ C_{j}=\operatorname{cap} \Omega_{j}\geq\operatorname{cap} \tilde{E}_{j} \geq A_{2} \lambda ^{j+1} \varphi^{n-3}\bigl(\lambda^{j+1}\bigr). $$

Note that

$$ \sum_{j=1}^{\infty} \frac{\lambda^{j} \varphi^{n-3}(\lambda^{j})}{\lambda ^{j(n-2)}}=\infty $$

is equivalent to

$$ \sum_{j=1}^{\infty} \frac{\lambda^{j+1} \varphi^{n-3}(\lambda ^{j+1})}{\lambda^{(j+1)(n-2)}}=\infty. $$

Hence, (1.5) holds if and only if

$$ \sum_{j=1}^{\infty} \frac{\lambda^{j} \varphi^{n-3}(\lambda^{j})}{\lambda ^{j(n-2)}}=\infty, $$

whose integral representation is exactly (1.6). Therefore, 0 is a regular point if and only if (1.6) holds and hence Theorem 1.1 is completed. □

3 Conclusion

Let Ω be a bounded domain in \(R^{n}\) and \(x_{0}\in\partial\Omega \). The Wiener criterion provides a necessary and sufficient condition on Ω to solve the Dirichlet problem of the Laplace equation (i.e., (1.1)). However, its geometric meaning is not clear and it is not easy to verify whether a domain satisfies the Wiener criterion at some boundary point. One interesting case is that Ω is constructed by a spine near \(x_{0}\) (see (1.2)). It is natural to ask what condition the spine should satisfy to guarantee that \(x_{0}\) is a regular point. When the dimension \(n=2\) or 3, the results are well known. In this paper, we consider \(n>3\) and prove that \(x_{0}\) is a regular point with respect to the Dirichlet problem if and only if the spine satisfies (1.6).