Abstract
By using the generalised Dirichlet integral inequality with continuous functions on the boundary of the upper half-space, we prove new types of solutions for the Neumann problem with fast-growing continuous data on the boundary. Given any harmonic function with its negative part satisfying similarly fast-growing conditions, we obtain weaker boundary integral condition.
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1 Introduction
Let \(\mathbf{R}^{n}\) denote the n-dimensional Euclidean space, where \(n\geq3\). We denote two points L and N in \(\mathbf{R}^{n}\) by \(L=(x',x_{n})\) and \(N=(y',y_{n})\), respectively, where \(x'=(x_{1},x_{2},\ldots,x_{n-1}) \), \(y'=(y_{1},y_{2},\ldots,y_{n-1}) \), \(x_{n} \in\mathbf{R}\) and \(y_{n} \in\mathbf{R}\). The Euclidean distance of them is denoted by \(\vert L-N\vert \). Let E be a subset of \(\mathbf{R}^{n}\), we denote the boundary and closure of it by ∂E and E̅, respectively.
The set
is denoted by \(\mathcal{T}_{n}\), which is called the upper half-space. Let F be a subset of \(\mathbf{R}_{+}\cup\{0\}\). Then two sets
are denoted by \(\mathcal{T}_{n}E\) and \(\partial\mathcal{T}_{n}E\), respectively.
Let \(B_{n}(r)\) denote the open ball with center at the origin and radius r, where \(r>0\). By \(S_{n}(r)\) we denote \(\mathcal{T}_{n}\cap\partial B_{n}(r)\). When g is a function defined by \(\sigma_{n}(r)=\mathcal{T}_{n}\cap B_{n}(r)\), the mean of g is defined by
where \(s_{n}\) is the surface area of \(B_{n}(1)\) and \(d\sigma_{L}\) is the surface element on \(B_{n}(r)\) at \(L\in\sigma_{n}(r)\).
Let \(h(L)\) be a function on \(\mathcal{T}_{n}\). In this paper we denote \(h^{+}=\max\{h,0\}\), \(h^{-}=-\min\{h,0\}\) and \([c]\) is the integer part of c, where \(c\in\mathbf {R}\). Let \(\partial/\partial n\) denote differentiation along the inward normal into \(\mathcal{T}_{n}\). We use the Lebesgue measure \(dL=dx'\,dx_{n}\), where \(dx'=dx_{1}\cdots\,dx_{n-1}\).
Let f be a continuous function on \(\partial\mathcal{T}_{n}\). If h is a harmonic function on \(\mathcal{T}_{n}\) and
then we say that h is a solution of the Neumann problem on \(\mathcal {T}_{n}\) with respect to f.
The uniqueness and the existence of solutions of the Neumann problem on \(\mathcal{T}_{n}\) with a continuous function on \(\partial\mathcal{T}_{n}\) were given by Su (see [1, 2]).
Theorem A
(see [3], Theorem 1)
Let \(f(N) \) (\(N=(y',0)\)) be a function continuous on \(\partial\mathcal{T}_{n}\) such that
Then the Neumann integral
is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f satisfying
as \(r\rightarrow+\infty\), where \(\rho_{n}=2\{(n-2)s_{n}\}^{-1}\).
Theorem B
(see [3], Theorem 3)
Let k be a positive integer, f be a continuous function on \(\partial \mathcal{T}_{n}\) such that (1.1) holds and \(h(L)\) be a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f satisfying
as \(r\rightarrow+\infty\). Then
for any \(L=(x',x_{n})\), where d is a constant, \(\Pi(x')\) is a polynomial of degree less than k on \(\partial\mathcal{T}_{n}\) and
Recently, Ren and Yang (see [4]) extended Theorems A and B by defining generalised Neumann integrals with continuous functions under less restricted conditions than (1.1). Meanwhile, they also proved that for any continuous function f on \(\partial\mathcal{T}_{n}\) there exists a solution of Neumann problem on \(\mathcal{T}_{n}\). To state them, we need some preliminaries.
Let L and N be two points on \(\mathcal{T}_{n}\) and \(\partial\mathcal {T}_{n}\), respectively. By \(\langle L,N\rangle\) we denote the usual inner product in \(\mathbf{R}^{n}\). We denote
where \(\vert N\vert >\vert L\vert \),
and \(G_{k,n}\) is the n-dimensional Legendre polynomial of degree k.
As in [2], we shall use the following generalised Dirichlet kernel. For a non-negative integer l, two points \(L\in\mathcal{T}_{n}\) and \(N\in\partial\mathcal{T}_{n}\), we put
The generalised Neumann kernel \(\mathbb{K}_{l,n}(L,N)\) on \(\mathcal {T}_{n}\) is defined by (see [2])
where \(L\in\mathcal{T}_{n}\), \(N\in\partial\mathcal{T}_{n}\) and
As for similar generalised Dirichlet kernel in a half plane and smooth cone, we refer the reader to the papers by Yang and Ren (see [5]), Zhao and Yamada (see [6]) and Su (see [1]).
Let \(f(N)\) be a continuous function on \(\partial\mathcal{T}_{n}\). Then the generalised Neumann integral on \(\mathcal{T}_{n}\) can be defined by
Ren and Yang proved the following results.
Theorem C
(see [4], Corollary 1)
Let \(1< p< \infty\), \(n+\beta-2>-(n-1)(p-1)\) and
Let \(f(N) \) (\(N=(y',0)\)) be a continuous function on \(\partial\mathcal {T}_{n}\) such that
Then the generalised Neumann integral \(\mathbb{H}_{l,n}[f](L)\) is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f satisfying
as \(r\rightarrow+\infty\).
Theorem D
(see [4], Theorem 3)
Let \(1\leq p< \infty\), \(\beta>1-p\), l be a positive integer and
Let \(f(N)\) be a continuous function on \(\partial\mathcal{T}_{n}\) satisfying (1.3). If \(h(L)\) is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f such that
then
for any \(L=(x',x_{n})\), where d is a constant, \(\Pi(x')\) is a polynomial of degree less than \(l+[1+\frac{\beta-1}{p}]\) on \(\partial \mathcal{T}_{n}\).
From Theorems A, B, C and D, it is easy to see that the continuous boundary function f grows slowly on \(\partial\mathcal{T}_{n}\). It is natural to ask what will happen if f is replaced by a fast-growing continuous function on \(\partial\mathcal{T}_{n}\). In this paper, we shall solve this problem and explicitly give a new solution of the Neumann problem on \(\partial\mathcal{T}_{n}\).
Define
where \(\tau(r)\) is a nondecreasing and continuously differentiable function satisfying \(\tau(r)\geq1\) for any \(r\in\mathbf{R}^{+}\cup\{0\}\).
From these we see that there is a sufficiently large positive number r such that for any \(t>r\)
where ϵ is a sufficiently small positive number satisfying \(\epsilon_{0}+\epsilon<1\).
Let \(\mathfrak{A}_{\varpi}\) be the set of continuous functions \(f(N)\) (\(N=(y',0)\)) on \(\partial\mathcal{T}_{n}\) satisfying
where ϖ is a real number such that \(\varpi>2\).
2 Results
Now we state our results.
Theorem 1
If \(f\in\mathfrak{A}_{\varpi}\), then generalised Neumann integral \(\mathbb{H}_{[\tau(\vert y'\vert )+\varpi],n}[f](L)\) is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f.
Then we shall prove that if the negative part of a harmonic function satisfies a fast-growing condition, then its positive part satisfies the similar condition. That is to say, the condition of Theorem 1 may be replaced by a weaker integral condition. To state this result, we also need some notations.
Let \(\mathfrak{B}_{\varpi}\) be the set of continuous functions \(f(N)\) (\(N=(y',y_{n})\)) on \(\mathcal{T}_{n}\) satisfying
By \(\mathfrak{C}_{\varpi}\) we denote the set of all continuous functions \(h(N)\) on \(\overline{\mathcal{T}_{n}}\), harmonic on \(\mathcal {T}_{n}\) with \(h^{-}(N)\in\mathfrak{B}_{\varpi}\) and \(h^{-}(y')\in\mathfrak {A}_{\varpi}\).
Theorem 2
The conclusion of Theorem 1 remains valid if its condition is replaced by \(h\in\mathfrak{C}_{\varpi}\).
Theorem 3
If \(h\in\mathfrak{C}_{\varpi}\), then there exists a harmonic function \(\Lambda(L)\) with normal derivative vanishes on \(\partial\mathcal{T}_{n}\) such that
where \(L\in\overline{\mathcal{T}}_{n}\).
3 Lemmas
Lemma 1
Let \(L\in\mathcal{T}_{n}\) and \(N\in\partial\mathcal{T}_{n}\) such that \(\vert N\vert \geq\max\{1,2\vert L\vert \}\). Then (see [7])
where M is a positive constant.
Lemma 2
Let \(\mathbb{W}(L,N)\) (\(N\in\partial \mathcal{T}_{n}\)) be a locally integrable function for any fixed point \(L\in\mathcal{T}_{n}\), \(g(N)\) be a upper semicontinuous and locally integrable function on \(\partial\mathcal{T}_{n}\). Set
for any \(N\in\partial\mathcal{T}_{n}\) and \(L\in\mathcal{T}_{n}\).
Suppose that the following two conditions hold:
-
(I)
There are a positive number R and a neighborhood \(B(N^{*})\) of \(N^{*}\) (\(\in\partial\mathcal{T}_{n}\)) satisfying
$$\int_{\partial\mathcal{T}_{n}[R,+\infty)\cup\partial\mathcal {T}_{n}(-\infty,-R]}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial}{\partial x_{n}}\mathbb {K}(L,N)\biggr\vert \,dN< \epsilon, $$where \(\epsilon> 0\).
-
(II)
There exists a positive number R satisfying
$$\limsup_{L\rightarrow N^{*},L\in\mathcal{T}_{n}} \int_{\partial\mathcal {T}_{n}(-R,R)}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial}{\partial x_{n}}\mathbb{W}(L,N)\biggr\vert \,dN=0 $$for any \(N^{*}\in\partial\mathcal{T}_{n}\).
Then
Proof
Let \(N^{*}\) be any point of \(\partial \mathcal{T}_{n}\) and ϵ be any positive number. There exists a positive number \(R^{*}\) satisfying
for any \(L=(x',x_{n})\in\mathcal{T}_{n}\cap B(N^{*})\) from (I).
Let ϕ be a continuous function on \(\partial\mathcal{T}_{n}\) such that \(0\leq\phi\leq1\) and
Let \(\mathbb{K}_{0,n}^{j}(L,N)\) be the Neumann function of \(\mathcal{T}_{n}(-j,j)\), where j is a positive integer. Since
on \(\mathcal{T}_{n}(-j,j)\) converges monotonically to 0 as \(j\rightarrow\infty\), we can find an integer \(j^{*}\) satisfying \(j^{*}>2R^{*}\) such that
for any \(L=(x',x_{n})\in B(N^{*})\cap\mathcal{T}_{n}\).
Then we have from (3.2) and (3.3) that
for any \(L=(x',x_{n})\in\mathcal{T}_{n}\cap B(N^{*})\).
Consider an upper semicontinuous function
on \(\partial\mathcal{T}_{n}(-j^{*},j^{*})\) and denote the Perron-Wiener-Brelot solution of the Neumann problem on \(\mathcal{T}_{n}(-j^{*},j^{*})\) by \(\mathbb{H}_{\psi}(L;\mathcal {T}_{n}(-j^{*},j^{*}))\). We know that
We also have
Hence we obtain
which together with (II) and (3.4) gives (3.1). □
Lemma 3
Let \(r>1\) and \(h(N)\) (\(N=(y',y_{n})\)) be a function harmonic on \(\mathcal {T}_{n}\). Then
where
and
4 Proof of Theorem 1
We have from (1.4)
for any \(k>k_{r}=[2r]+1\), where \(M_{1}(r)\) is a positive constant dependent only on r.
We have for any \(L\in\mathcal{T}_{n}\) and \(\vert L\vert \leq R\)
from Lemma 1 and (1.5). So \(\mathbb{H}_{[\tau(\vert y'\vert )+\varpi ],n}(L)\) is absolutely convergent.
Next we shall prove that
for any \(N'=(y',0)\in\partial\mathcal{T}_{n}\). By applying Lemma 2 to \(-g(y')\) and \(g(y')\) by setting
then we shall see that (I) and (II) hold. Take any \(N'=(y',0)\in\partial\mathcal{T}_{n}\) and any \(\epsilon>0\). There exists a number R (\({>}\max\{2(\delta+y'),1\}\)) satisfying
for any \(L\in\mathcal{T}_{n} \cap U(N',\delta)\) from (1.5) and (4.2), which is (I) in Lemma 2. To see (II), we only need to observe from (1.2) that for any \(N'\in\partial\mathcal{T}_{n}\)
So Theorem 1 is proved.
5 Proof of Theorem 2
Lemma 2 gives
where
Since \(h\in\mathfrak{C}_{\varpi}\), we obtain by (2.1)
We have by (1.5)
From (5.1), (5.2) and Lemma 2, we see that
Set
It is easy to see that
from (1.4), which shows that
for any \(\vert y'\vert \geq1\), where \(M_{2}\) is a positive constant.
It follows that
from (5.3).
Then Theorem 2 is proved from \(\vert h\vert =h^{+}+h^{-}\).
6 Proof of Theorem 3
Put \(h'(L)= h(L)-\mathbb{H}_{[\tau(\vert y'\vert )+\varpi],n}(L)\). Then it is easy to see that \(h'(L)\) is harmonic on \(\mathcal{T}_{n}\) with normal derivative vanishes on \(\partial\mathcal{T}_{n} \) and \(h'(L)\) can be continuously extended to \(\overline{\mathcal{T}_{n}}\). By applying the Schwarz reflection principle [8], p.68, to \(h'(L)\), it follows that there is a function harmonic on \(\mathcal{T}_{n}\) satisfying \(h(L^{*})=-h'(L)=-(h(L)-\mathbb{H}_{[\tau(\vert y'\vert )+\varpi],n}(L))\) for \(L\in \overline{T}_{n}\), where ∗ denotes reflection in \(\partial\mathcal{T}_{n}\) just as \(L^{*}=(x', -x_{n})\). Thus \(h(L)=\Lambda(L)+\mathbb{H}_{[\tau(|y'|)+\varpi],n}(L)\) for all \(L \in \overline{\mathcal{T}}_{n} \), where \(\Lambda(L)\) is a harmonic function on \(\mathcal{T}_{n}\) with normal derivative which vanishes continuously on \(\partial\mathcal{T}_{n}\). Theorem 3 is proved.
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Acknowledgements
This work was supported by the Natural Science Foundation of Zhejiang Province (No. LQ13A010019). The authors would like to thank the referee for invaluable comments and insightful suggestions.
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Li, W., Zaprawa, M.A. Applications of the generalised Dirichlet integral inequality to the Neumann problem with fast-growing continuous data. J Inequal Appl 2016, 250 (2016). https://doi.org/10.1186/s13660-016-1185-y
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DOI: https://doi.org/10.1186/s13660-016-1185-y