The following result is an augmented Riesz decomposition method, which was used to study the boundary behaviors of Poisson integral. For similar results for solutions of the equilibrium equations with angular velocity, we refer the reader to the paper by Wang et al. (see [3]).
Lemma 1
For
\(W'\in\partial{\mathcal{T}_{n}(\Gamma)}\)and
\(\epsilon>0\), there exist a positive numberRand a neighborhood
\(B(W')\)of
\(W'\)such that
$$ \frac{1}{c_{n}} \int_{S_{n}(\Gamma;(R,\infty))}\bigl\vert g(W)\bigr\vert \bigl\vert \operatorname{PI}_{\Gamma }(V,W)\bigr\vert \, d\sigma_{W}< \epsilon $$
(7)
for any
\(V=(r,\Lambda)\in\mathcal{T}_{n}(\Gamma)\cap B(W')\), wheregis an upper semi-continuous function. Then
$$ \limsup_{V\in\mathcal{T}_{n}(\Gamma), V\rightarrow W'}\operatorname{PI}_{\Gamma}[g](V) \leq g\bigl(W'\bigr). $$
(8)
Proof
Let \(W'=(l',\Phi')\) be any point of \(\partial{\mathcal{T}_{n}(\Gamma)}\) and ϵ (>0) be any number. There exists a positive number \(R'\) satisfying
$$ \frac{1}{c_{n}} \int_{S_{n}(\Gamma;(R',\infty))}\bigl\vert \operatorname{PI}_{\Gamma }(V,W) \bigr\vert \bigl\vert g(W)\bigr\vert \, d\sigma_{W}\leq \frac{\epsilon}{4} $$
(9)
for any \(V=(r,\Lambda)\in\mathcal{T}_{n}(\Gamma)\cap B(W')\) from (7).
Let ϕ be continuous on \(\partial{\mathcal{T}_{n}(\Gamma)}\) such that \(0\leq\phi\leq1\) and
$$\phi=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 & \mbox{on } S_{n}(\Gamma;(0,R'])\cup\{O\}, \\ 0 & \mbox{on } S_{n}(\Gamma;(2R',\infty)). \end{array}\displaystyle \right . $$
Let \(\mathrm{G}_{\mathcal{T}_{n}(\Gamma;(0,j))}\) be a Green’s function on \(\mathcal{T}_{n}(\Gamma;(0,j))\), where j is a positive integer. Since \(\Gamma_{j}(V,W)=\mathrm{G}_{\mathcal{T}_{n}(\Gamma)}(V,W)-\mathrm {G}_{\mathcal{T}_{n}(\Gamma;(0,j))}(V,W)\) on \(\mathcal{T}_{n}(\Gamma;(0,j))\) converges monotonically to 0 as \(j\rightarrow\infty\). Then we can find an integer \(j'\), \(j'>2R'\) such that
$$ \frac{1}{c_{n}} \int_{S_{n}(\Gamma;(0,2R'))}\biggl\vert \frac{\partial }{\partial n_{W}}\Gamma_{j'}(V,W) \biggr\vert \bigl\vert \phi(W)g(W)\bigr\vert \, d\sigma_{W}< \frac{\epsilon}{4} $$
(10)
for any \(V=(r,\Lambda)\in\mathcal{T}_{n}(\Gamma)\cap B(W')\).
Then we have from (9) and (10)
$$\begin{aligned}& \frac{1}{c_{n}} \int_{\partial{\mathcal{T}_{n}(\Gamma)}}\operatorname {PI}_{\Gamma}(V,W)g(W)\,d \sigma_{W} \\& \quad \leq \frac{1}{c_{n}} \int_{S_{n}(\Gamma;(0,2R'))}\frac{\partial \mathrm{G}_{\mathcal{T}_{n}(\Gamma;(0,j'))}(V,W) }{\partial n_{W}}\phi(W)g(W)\,d\sigma_{W} \\& \qquad {} + \frac{1}{c_{n}} \int_{S_{n}(\Gamma;(0,2R'))}\bigl\vert \phi(W)g(W)\bigr\vert \biggl\vert \frac {\partial \Gamma_{j'}(V,W)}{\partial n_{W}}\biggr\vert \,d\sigma_{W} \\& \qquad {} + \frac{2}{c_{n}} \int_{S_{n}(\Gamma;(R',\infty))}\bigl\vert \operatorname {PI}_{\Gamma}(V,W) \bigr\vert \bigl\vert g(W)\bigr\vert \,d\sigma_{W} \\& \quad \leq \frac{1}{c_{n}} \int_{S_{n}(\Gamma;(0,2R'))}\frac{\partial \mathrm{G}_{\mathcal{T}_{n}(\Gamma;(0,j'))}(V,W) }{\partial n_{W}}\phi(W)g(W)\,d\sigma_{W}+ \frac{3}{4}\epsilon \end{aligned}$$
(11)
for any \(V=(r,\Lambda)\in\mathcal{T}_{n}(\Gamma)\cap B(W')\).
Consider an upper semi-continuous function
$$\eta(W)=\left \{ \textstyle\begin{array}{l@{\quad}l} \phi(W)g(W) & \mbox{on } S_{n}(\Gamma;(0,2R'])\cup\{O\}, \\ 0 & \mbox{on } \partial{\mathcal{T}_{n}(\Gamma;(0,j'))}-S_{n}(\Gamma ;(0,2R'])-\{O\}, \end{array}\displaystyle \right . $$
on \(\partial{\mathcal{T}_{n}(\Gamma;[0,j'))}\) and denote the PWB solution of the Dirichlet problem on \(\mathcal{T}_{n}(\Gamma;(0,j'))\) by \(H_{\eta}(P;\mathcal{T}_{n}(\Gamma ;(0,j')))\) (see, e.g., [4]); we know that
$$ \frac{1}{c_{n}} \int_{S_{n}(\Gamma;(0,2R'))}\frac{\partial \mathrm{G}_{\mathcal{T}_{n}(\Gamma;(0,j'))}(V,W) }{\partial n_{W}}\phi(W)g(W)\, d\sigma_{W}=H_{\eta} \bigl(P;\mathcal{T}_{n}\bigl(\Gamma;\bigl(0,j'\bigr)\bigr) \bigr) $$
(12)
(see [5], Theorem 3). If \(\mathcal{T}_{n}(\Gamma;(0,j'))\) is not a Lipschitz domain at O, we can prove (12) by considering a sequence of the Lipschitz domains \(\mathcal{T}_{n}(\Gamma;(\frac{1}{m},j'))\) which converges to \(\mathcal{T}_{n}(\Gamma;(0,j'))\) as \(m\rightarrow\infty\). We also have
$$\limsup_{V\in\mathcal{T}_{n}(\Gamma), V\rightarrow W'}H_{\eta }\bigl(P;\mathcal{T}_{n} \bigl(\Gamma;\bigl(0,j'\bigr)\bigr)\bigr)\leq\limsup _{Q\in S_{n}(\Gamma), Q\rightarrow W'}\eta(W)=g\bigl(W'\bigr) $$
(see, e.g., [4], Lemma 8.20). Hence we know that
$$\limsup_{V\in\mathcal{T}_{n}(\Gamma), V\rightarrow W'}\frac {1}{c_{n}} \int_{S_{n}(\Gamma;(0,2R'))}\phi(W)\frac{\partial \mathrm{G}_{\mathcal{T}_{n}(\Gamma;(0,j'))}(V,W) }{\partial n_{W}}g(W)\, d\sigma\leq g \bigl(W'\bigr). $$
With (11) this gives (8). □
The following growth properties play important roles in our discussions.
Lemma 2
(see [6])
Let
\(V=(r,\Lambda )\in\mathcal{T}_{n}(\Gamma)\)and
\(W=(t,\Phi)\in S_{n}(\Gamma)\). Then we have
$$\operatorname{PI}_{\Gamma}(V,W)\leq M r^{\varrho^{-}}t^{\varrho^{+}-1} \eta(\Lambda)\quad \biggl(0< \frac{t}{r}\leq \frac{4}{5}\biggr) $$
and
$$\operatorname{PI}_{\Gamma}(V,W)\leq M r^{\varrho^{+}}t^{\varrho^{-}-1} \eta(\Lambda) \quad \biggl(0< \frac{r}{t}\leq \frac{4}{5}\biggr). $$
Let
\(V=(r,\Lambda)\in\mathcal{T}_{n}(\Gamma)\)and
\(W=(t,\Phi)\in S_{n}(\Gamma; (\frac{4}{5}r,\frac{5}{4}r))\). Then we have
$$\operatorname{PI}_{\Gamma}(V,W)\leq M\frac{\eta(\Lambda)}{t^{n-1}}+M \frac{r\eta(\Lambda)}{|P-Q|^{n}}. $$
Let
\(\mathrm{G}_{\mathcal{T}_{n}(\Gamma;(t_{1},t_{2}))}\)be the Green’s function of
\(\mathcal{T}_{n}(\Gamma;(t_{1},t_{2}))\). Then we have
$$\frac{\partial\mathrm{G}_{\mathcal{T}_{n}(\Gamma ;(t_{1},t_{2}))}((t_{1},\Phi),(r,\Lambda))}{\partial t}\leq M\biggl(\frac{t_{1}}{r}\biggr)^{-\varrho^{-}} \frac{\eta(\Phi)\eta(\Lambda)}{t_{1}^{n-1}} $$
and
$$-M\biggl(\frac{r}{t_{2}}\biggr)^{\varrho^{+}}\frac{\eta(\Phi)\eta(\Lambda )}{t_{2}^{n-1}}\leq \frac{\partial\mathrm{G}_{\mathcal{T}_{n}(\Gamma ;(t_{1},t_{2}))}((t_{2},\Phi),(r,\Lambda))}{\partial t}, $$
where
\(0<2t_{1}<r<\frac{1}{2}t_{2}<+\infty\).
Many previous studies (see [7, 8]) focused on the following lemma with respect to the half space and its applications.
Lemma 3
(see [2], Lemma 2)
Ifhis a function harmonic in a domain containing
\(\mathcal{T}_{n}(\Gamma ;(1,R))\), where
\(R>1\), then we have
$$\lambda \int_{S_{n}(\Gamma;R)}h\eta R^{\varrho^{-}-1} d S_{R} + \int_{S_{n}(\Gamma;(1,R))}h \bigl(t^{\varrho^{-}}-t^{\varrho ^{+}}R^{-\lambda} \bigr){\partial\eta}/{\partial n} \, d\sigma_{W}+d_{1}+ \frac{d_{2}}{R^{\lambda}}=0, $$
where
$$d_{1}= \int_{S_{n}(\Gamma;1)}\varrho^{-}h\eta-\eta({\partial h}/{\partial n}) \, dS_{1} $$
and
$$d_{2}= \int_{S_{n}(\Gamma;1)}\eta({\partial h}/{\partial n})-\varrho ^{+}h\eta \, dS_{1}. $$