RETRACTED ARTICLE: Some new results on the boundary behaviors of harmonic functions with integral boundary conditions

Abstract

In this paper, using a generalized Carleman formula, we prove two new results on the boundary behaviors of harmonic functions with integral boundary conditions in a smooth cone, which generalize some recent results.

1 Introduction

Let $\mathbf{R}^{n}$ ($n\geq2$) be the n-dimensional Euclidean space. A point in $\mathbf{R}^{n}$ is denoted by $V=(X,y)$, where $X=(x_{1},x_{2},\ldots,x_{n-1})$. The boundary and the closure of a set E in $\mathbf{R}^{n}$ are denoted by ∂E and E̅, respectively.

We introduce a system of spherical coordinates $(l,\Lambda)$, $\Lambda=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})$, in $\mathbf{R}^{n}$ that are related to Cartesian coordinates $(x_{1},x_{2},\ldots,x_{n-1},y)$ by $y=l\cos\theta_{1}$.

The unit sphere and the upper half unit sphere in $\mathbf{R}^{n}$ are denoted by $\mathbf{S}^{n-1}$ and $\mathbf{S}_{+}^{n-1}$, respectively. For simplicity, a point $(1,\Lambda)$ on $\mathbf{S}^{n-1}$ and the set $\{\Lambda; (1,\Lambda)\in\Gamma\}$ for a set $\Gamma\subset\mathbf{S}^{n-1}$ are often identified with Λ and Γ, respectively. For two sets $\Xi\subset\mathbf{R}_{+}$ and $\Gamma\subset \mathbf{S}^{n-1}$, the set $\{(l,\Lambda)\in\mathbf{R}^{n}; l\in\Xi,(1,\Lambda)\in\Gamma\}$ in $\mathbf{R}^{n}$ is simply denoted by $\Xi\times\Gamma$.

We denote the set $\mathbf{R}_{+}\times\Gamma$ in $\mathbf{R}^{n}$ with the domain Γ on $\mathbf{S}^{n-1}$ by $T_{n}(\Gamma)$. We call it a cone. In particular, the half-space $\mathbf{R}_{+}\times\mathbf{S}_{+}^{n-1}$ is denoted by $T_{n}(\mathbf{S}_{+}^{n-1})$. The sets $I\times\Gamma$ and $I\times\partial{\Gamma}$ with an interval on R are denoted by $T_{n}(\Gamma;I)$ and $\mathcal{S}_{n}(\Gamma;I)$, respectively. We denote $T_{n}(\Gamma)\cap S_{l}$ by $\mathcal{S}_{n}(\Gamma ; l)$, and we denote $\mathcal{S}_{n}(\Gamma; (0,+\infty))$ by $\mathcal{S}_{n}(\Gamma)$.

The ordinary Poisson in $T_{n}(\Gamma)$ is defined by

$$c_{n}\mathbb{PI}_{\Gamma}(V,W)=\frac{\partial\mathbb{G}_{\Gamma }(V,W)}{\partial n_{W}}, $$

where ${\partial}/{\partial n_{W}}$ denotes the differentiation at W along the inward normal into $T_{n}(\Gamma)$, and $\mathbb{G}_{\Gamma }(V,W)$ ($P, Q\in T_{n}(\Gamma)$) is the Green function in $T_{n}(\Gamma)$. Here, $c_{2}=2$ and $c_{n}=(n-2)w_{n}$ for $n\geq3$, where $w_{n}$ is the surface area of $\mathbf{S}^{n-1}$.

Let $\Delta_{n}^{*}$ be the spherical part of the Laplace operator, and Γ be a domain on $\mathbf{S}^{n-1}$ with smooth boundary ∂Γ. Consider the Dirichlet problem (see [1])

$$\begin{aligned}& \bigl(\Delta_{n}^{*}+\tau \bigr)\psi=0 \quad \mbox{on } \Gamma, \\& \psi=0 \quad \mbox{on } \partial{\Gamma}. \end{aligned}$$

We denote the least positive eigenvalue of this boundary problem by τ and the normalized positive eigenfunction corresponding to τ by $\psi(\Lambda)$. In the sequel, for brevity, we shall write χ instead of $\aleph^{+}-\aleph^{-}$, where

$$2\aleph^{\pm}=-n+2\pm\sqrt{(n-2)^{2}+4\tau}. $$

The estimate we deal with has a long history tracing back to known Matsaev’s estimate of harmonic functions from below in the half-plane (see, e.g., Levin [2], p.209).

Theorem A

Let $A_{1}$be a constant, and let $h(z)$ ($|z|=R$) be harmonic on $T_{2}(\mathbf{S}_{+}^{1})$and continuous on $\overline{T_{2}(\mathbf{S}_{+}^{1})}$. Suppose that

$$h(z)\leq A_{1}R^{\rho},\quad z\in T_{2} \bigl( \mathbf{S}_{+}^{1} \bigr), R>1, \rho>1, $$

and

$$\bigl|h(z)\bigr|\leq A_{1}, \quad R\leq1, z\in\overline{T_{2} \bigl( \mathbf{S}_{+}^{1} \bigr)}. $$

Then

$$h(z)\geq-A_{1}A_{2} \bigl(1+R^{\rho}\bigr) \sin^{-1}\alpha, $$

where $z=Re^{i\alpha}\in T_{2}(\mathbf{S}_{+}^{1})$, and $A_{2}$is a constant independent of $A_{1}$, R, α, and the function $h(z)$.

In 2014, Xu and Zhou [3] considered Theorem A in the half-space. Pan et al. [4], Theorems 1.2 and 1.4, obtained similar results, slightly different from the following Theorem B.

Theorem B

Let $A_{3}$be a constant, and $h(V)$ ($\vert V\vert =R$) be harmonic on $T_{n}(\mathbf{S}_{+}^{n-1})$and continuous on $\overline{T_{n}(\mathbf{S}_{+}^{n-1})}$. If

$$ h(V)\leq A_{3}R^{\rho},\quad P\in T_{n} \bigl( \mathbf {S}_{+}^{n-1} \bigr), R>1, \rho>n-1, $$

(1.1)

and

$$ \bigl\vert h(V)\bigr\vert \leq A_{3}, \quad R\leq1, P\in \overline{T_{n} \bigl( \mathbf {S}_{+}^{n-1} \bigr)}, $$

(1.2)

then

$$h(V)\geq-A_{3}A_{4} \bigl(1+R^{\rho} \bigr) \cos^{1-n}\theta_{1}, $$

where $V\in T_{n}(\mathbf{S}_{+}^{n-1})$, and $A_{4}$is a constant independent of $A_{3}$, R, $\theta_{1}$, and the function $h(V)$.

Recently, Pang and Ychussie [5], Theorem 1, further extended Theorems A and B and proved Matsaev’s estimates for harmonic functions in a smooth cone.

Theorem C

LetKbe a constant, and $h(V)$ ($V=(R,\Lambda)$) be harmonic on $T_{n}(\Gamma)$and continuous on $\overline{T_{n}(\Gamma)}$. If

$$ h(V)\leq KR^{\rho(R)}, \quad V=(R,\Lambda)\in T_{n} \bigl(\Gamma;(1,\infty) \bigr),\quad \rho(R)> \aleph^{+}, $$

(1.3)

and

$$ h(V)\geq-K, \quad R\leq1, \quad V=(R,\Lambda) \in \overline{T_{n}(\Gamma)}, $$

(1.4)

then

$$h(V)\geq-KM \biggl(1+ \biggl(\frac{N+1}{N}R \biggr)^{\rho(\frac{N+1}{N}R)} \biggr) \psi ^{1-n}(\Lambda), $$

where $V\in T_{n}(\Gamma)$, N (≥1) is a sufficiently large number, andMis a constant independent ofK, R, $\psi(\Lambda)$, and the function $h(V)$.

In this paper, we obtain two new results on the lower bounds of harmonic functions with integral boundary conditions in a smooth cone (Theorems 1 and 2), which further extend Theorems A, B, and C. Our proofs are essentially based on the Riesz decomposition theorem (see [6]) and a modified Carleman formula for harmonic functions in a smooth cone (see [5], Lemma 1).

In order to avoid complexity of our proofs, we assume that $n\geq3$. However, our results in this paper are also true for $n=2$. We use the standard notations $h^{+}=\max\{h,0\}$ and $h^{-}=-\min\{h,0\}$. All constants appearing further in expressions will be always denoted M because we do not need to specify them. We will always assume that $\eta(t)$ and $\rho(t)$ are nondecreasing real-valued functions on an interval $[1,+\infty)$ and $\rho(t)> \aleph^{+}$ for any $t\in[1,+\infty)$.

2 Main results

First of all, we shall state the following result, which further extends Theorem C under weak boundary integral conditions.

Theorem 1

Let $h(V)$ ($V=(R,\Lambda)$) be harmonic on $T_{n}(\Gamma)$and continuous on $\overline{T_{n}(\Gamma)}$.

Suppose that the following conditions (I) and (II) are satisfied:

• (I) For any $V=(R,\Lambda)\in T_{n}(\Gamma;(1,\infty))$, we have

  $$ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-}t^{\aleph^{-}}{\partial\psi }/{ \partial n}\,d\sigma_{W} \leq M\eta(R) (cR)^{\rho(cR)-\aleph^{+}} $$

  (2.1)

  and

  $$ \chi \int_{\mathcal{S}_{n}(\Gamma ;R)}h^{-}R^{\aleph^{-}-1}\psi d S_{R} \leq M\eta(R) (cR)^{\rho(cR)-\aleph^{+}}. $$

  (2.2)

• (II) For any $V=(R,\Lambda)\in T_{n}(\Gamma;(0,1])$, we have

  $$ h(V)\geq-\eta(R). $$

  (2.3)

  Then

  $$h(V)\geq-M\eta(R) \bigl(1+(cR)^{\rho(cR)} \bigr)\psi^{1-n}( \Lambda), $$

  where $V\in T_{n}(\Gamma)$, N (≥1) is a sufficiently large number, andMis a constant independent ofR, $\psi(\Lambda)$, and the functions $\eta(R)$and $h(V)$.

Remark 1

From the proof of Theorem 1 it is easy to see that condition (I) in Theorem 1 is weaker than that in Theorem C in the case $c\equiv(N+1)/{N}$ and $\eta (R)\equiv K$, where N (≥1) is a sufficiently large number, and K is a constant.

Theorem 2

The conclusion of Theorem  1remains valid if (I) in Theorem  1is replaced by

$$ h(V)\leq\eta(R)R^{\rho(R)}, \quad V=(R,\Lambda)\in T_{n} \bigl(\Gamma;(1,\infty) \bigr). $$

(2.4)

Remark 2

In the case $c\equiv(N+1)/{N}$ and $\eta(R)\equiv K$, where N (≥1) is a sufficiently large number and K is a constant, Theorem 2 reduces to Theorem C.

3 Proof of Theorem 1

By the Riesz decomposition theorem (see [6]) we have

$$ -h(V)= \int_{\mathcal{S}_{n}(\Gamma;(0,R))}\mathcal{PI}_{\Gamma }(V,W)-h(W)\,d\sigma_{W}+ \int_{\mathcal{S}_{n}(\Gamma;R)}\frac{\partial \mathbb{G}_{\Gamma,R}(V,W)}{\partial R}-h(W)\,dS_{R}, $$

(3.1)

where $V=(l,\Lambda)\in T_{n}(\Gamma;(0,R))$.

We next distinguish three cases.

Case 1. $V=(l,\Lambda)\in T_{n}(\Gamma;({5}/{4},\infty ))$ and $R={5l}/{4}$.

Since $-h(V)\leq h^{-}(V)$, we have

$$ -h(V)=\sum_{i=1}^{4} U_{i}(V) $$

(3.2)

from (3.1), where

$$\begin{aligned}& U_{1}(V)= \int_{\mathcal{S}_{n}(\Gamma;(0,1])}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}, \\& U_{2}(V)= \int_{\mathcal{S}_{n}(\Gamma;(1,{4l}/{5}])}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}, \\& U_{3}(V)= \int_{\mathcal{S}_{n}(\Gamma;({4l}/{5},R))}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}, \end{aligned}$$

and

$$U_{4}(V)= \int_{\mathcal{S}_{n}(\Gamma;R)}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}. $$

We have the following estimates:

$$ U_{1}(V)\leq M\eta(R)\psi(\Lambda) $$

(3.3)

and

$$ U_{2}(V) \leq M\eta(R) (cR)^{\rho(cR)}\psi(\Lambda) $$

(3.4)

from [7, 8] and (2.1).

We consider the inequality

$$ U_{3}(V)\leq U_{31}(V)+U_{32}(V), $$

(3.5)

where

$$U_{31}(V)=M \int_{\mathcal{S}_{n}(\Gamma;({4l}/{5},R))}\frac{-h(W) \psi(\Lambda)}{t^{n-1}}\frac{\partial\phi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W} $$

and

$$U_{32}(V)=Mr\psi(\Lambda) \int_{\mathcal{S}_{n}(\Gamma;({4l}/{5},R))}\frac{-h(W) l\psi(\Lambda)}{\vert V-W\vert ^{n}} \frac{\partial\phi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W}. $$

We first have

$$ U_{31}(V) \leq M\eta(R) (cR)^{\rho(cR)}\psi( \Lambda) $$

(3.6)

from (2.1).

We shall estimate $U_{32}(V)$. Take a sufficiently small positive number d such that

$$\mathcal{S}_{n} \bigl(\Gamma;({4l}/{5},R) \bigr)\subset B(P,{l}/{2}) $$

for any $V=(l,\Lambda)\in\Pi(d)$, where

$$\Pi(d)= \Bigl\{ V=(l,\Lambda)\in T_{n}(\Gamma); \inf _{(1,z)\in\partial\Gamma}\bigl\vert (1,\Lambda)-(1,z)\bigr\vert < d, 0< r< \infty \Bigr\} , $$

and divide $T_{n}(\Gamma)$ into two sets $\Pi(d)$ and $T_{n}(\Gamma)-\Pi(d)$.

If $V=(l,\Lambda)\in T_{n}(\Gamma)-\Pi(d)$, then there exists a positive $d'$ such that $\vert V-W\vert \geq{d}'l$ for any $Q\in \mathcal{S}_{n}(\Gamma)$, and hence

$$ U_{32}(V) \leq M\eta(R) (cR)^{\rho(cR)}\psi( \Lambda), $$

(3.7)

which is similar to the estimate of $U_{31}(V)$.

We shall consider the case $V=(l,\Lambda)\in\Pi(d)$. Now put

$$H_{i}(V)= \bigl\{ W\in\mathcal{S}_{n} \bigl( \Gamma;({4l}/{5},R) \bigr); 2^{i-1}\delta(V) \leq \vert V-W\vert < 2^{i} \delta(V) \bigr\} , $$

where

$$\delta(V)=\inf_{Q\in \partial{T_{n}(\Gamma)}}\vert V-W\vert . $$

Since $\mathcal{S}_{n}(\Gamma)\cap\{W\in\mathbf{R}^{n}: \vert V-W\vert < \delta (V)\}=\emptyset$, we have

$$U_{32}(V)=M\sum_{i=1}^{i(V)} \int_{H_{i}(V)}\frac{-h(W)r\psi(\Lambda)}{\vert V-W\vert ^{n}}\frac {\partial \psi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W}, $$

where $i(V)$ is a positive integer satisfying

$$2^{i(V)-1}\delta(V)\leq\frac{r}{2}< 2^{i(V)}\delta(V). $$

Since $r\psi(\Lambda)\leq M\delta(V)$ ($V=(l,\Lambda)\in T_{n}(\Gamma)$), similarly to the estimate of $U_{31}(V)$, we obtain

$$\int_{H_{i}(V)}\frac{-h(W)r\psi(\Lambda)}{\vert V-W\vert ^{n}}\frac{\partial \psi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W}\leq M\eta(R) (cR)^{\rho(cR)}\psi^{1-n}(\Lambda) $$

for $i=0,1,2,\ldots,i(V)$.

So

$$ U_{32}(V)\leq M\eta(R) (cR)^{\rho(cR)} \psi^{1-n}(\Lambda). $$

(3.8)

From (3.5), (3.6), (3.7), and (3.8) we see that

$$ U_{3}(V)\leq M\eta(R) (cR)^{\rho(cR)} \psi^{1-n}(\Lambda). $$

(3.9)

On the other hand, we have from (2.2) that

$$ U_{4}(V) \leq M\eta(R)R^{\rho(cR)}\psi(\Lambda). $$

(3.10)

We thus obtain from (3.3), (3.4), (3.9), and (3.10) that

$$ -h(V)\leq M\eta(R) \bigl(1+(cR)^{\rho(cR)} \bigr) \psi^{1-n}(\Lambda). $$

(3.11)

Case 2. $V=(l,\Lambda)\in T_{n}(\Gamma;({4}/{5},{5}/{4}])$ and $R={5l}/{4}$.

It follows from (3.1) that

$$-h(V)= U_{1}(V)+U_{5}(V)+U_{4}(V), $$

where $U_{1}(V)$ and $U_{4}(V)$ are defined as in Case 1, and

$$U_{5}(V)= \int_{\mathcal{S}_{n}(\Gamma;(1,R))}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}. $$

Similarly to the estimate of $U_{3}(V)$ in Case 1, we have

$$U_{5}(V)\leq M\eta(R) (cR)^{\rho(cR)}\psi^{1-n}( \Lambda), $$

which, together with (3.3) and (3.10), gives (3.11).

Case 3. $V=(l,\Lambda)\in T_{n}(\Gamma;(0,{4}/{5}])$.

It is evident from (2.3) that

$$-h\leq\eta(R), $$

which also gives (3.11).

Finally, from (3.11) we have

$$h(V)\geq -\eta(R)M \bigl(1+(cR)^{\rho(cR)} \bigr)\psi^{1-n}( \Lambda), $$

which is the conclusion of Theorem 1.

4 Proof of Theorem 2

We first apply a new type of Carleman’s formula for harmonic functions (see [5], Lemma 1) to $h=h^{+}-h^{-}$ and obtain

$$\begin{aligned} &\chi \int_{\mathcal{S}_{n}(\Gamma;R)}h^{+}R^{\aleph^{-}-1}\psi d S_{R} \\ &\qquad{}+ \int _{\mathcal{S}_{n}(\Gamma;(1,R))}h^{+} \bigl(t^{\aleph^{-}}-t^{\aleph ^{+}}R^{-\chi} \bigr) {\partial\psi}/{\partial n}\,d\sigma_{W}+d_{1}+d_{2}R^{-\chi} \\ &\quad=\chi \int_{\mathcal{S}_{n}(\Gamma;R)}h^{-}R^{\aleph^{-}-1}\psi d S_{R}+ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-} \bigl(t^{\aleph ^{-}}-t^{\aleph^{+}}R^{-\chi} \bigr) {\partial\psi}/{\partial n}\,d\sigma_{W}, \end{aligned}$$

(4.1)

where $dS_{R}$ denotes the $(n-1)$-dimensional volume elements induced by the Euclidean metric on $S_{R}$, and ${\partial}/{\partial n}$ denotes differentiation along the interior normal.

It is easy to see that

$$ \chi \int_{\mathcal{S}_{n}(\Gamma;R)}h^{+}R^{\aleph^{-}-1}\psi d S_{R} \leq M\eta(R) (cR)^{\rho(cR)-\aleph^{+}} $$

(4.2)

and

$$ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{+} \bigl(t^{\aleph^{-}}-t^{\aleph ^{+}}R^{-\chi} \bigr){\partial\psi}/{\partial n}\,d\sigma_{W}\leq M\eta(R) (cR)^{\rho (cR)-\aleph^{+}} $$

(4.3)

from (2.4).

We remark that

$$ d_{1}+d_{2}R^{-\chi} \leq M\eta(R) (cR)^{\rho(cR)-\aleph^{+}}. $$

(4.4)

We have (2.2) and

$$ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-} \bigl(t^{\aleph^{-}}-t^{\aleph^{+}}R^{-\chi} \bigr) {\partial\psi}/{\partial n}\,d\sigma_{W} \leq M\eta(R) (cR)^{\rho(cR)-\aleph^{+}}. $$

(4.5)

from (4.1), (4.2), (4.3), and (4.4).

Hence, (4.5) gives (2.1), which, together with Theorem 1, gives Theorem 2.