Abstract
As an application of an inequality for modified Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.
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1 Introduction and results
Let \(\mathbf{R}^{n}\) (\(n\geq2\)) be the n-dimensional Euclidean space. The boundary and the closure of a set E in \(\mathbf{R}^{n}\) are denoted by ∂E and E̅, respectively. The Euclidean distance of two points P and Q in \(\mathbf{R}^{n}\) is denoted by \(\vert P-Q\vert \). Especially, \(\vert P\vert \) denotes the distance of two points P and O in \(\mathbf{R}^{n}\), where O is the origin in \(\mathbf{R}^{n}\).
We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
Let \(B(P,r)\) denote the open ball with center at P and radius r (>0) in \(\mathbf{R}^{n}\). 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. The surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{-1}\) of \(\mathbf{S}^{n-1}\) is denoted \(w_{n}\). Let \(\Omega\subset\mathbf{S}^{n-1}\), a point \((1,\Theta)\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) are denoted Θ and Ω, respectively. For two sets \(\Lambda\subset\mathbf{R}_{+}\) and \(\Omega\subset\mathbf{S}^{n-1}\), we denote \(\Lambda\times\Omega=\{ (r,\Theta)\in\mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Omega\}\), where \(\mathbf{R}_{+}\) is the set of all positive real numbers.
For the set \(\Omega\subset\mathbf{S}^{n-1}\), we denote the set \(\mathbf{ R}_{+}\times\Omega\) in \(\mathbf{R}^{n}\) by \(C_{n}(\Omega)\), which is called a cone. For the set \(I\subset \mathbf{R}\), the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\) are denoted \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\), respectively, where R is the set of all real numbers. Especially, the set \(S_{n}(\Omega; \mathbf{R}_{+})\) is denoted \(S_{n}(\Omega)\).
Given a continuous function f on \(S_{n}(\Omega)\), we say that h is a solution of the Dirichlet problem in \(C_{n}(\Omega)\) with f, if h is a harmonic function in \(C_{n}(\Omega)\) and
Let \(\Omega\subset\mathbf{S}^{n-1}\) and \(\Delta^{*}\) be a Laplace-Beltrami on the unit sphere. Consider the Dirichlet problem (see, e.g. [2], p.41)
We denote the non-decreasing sequence of positive eigenvalues of it, repeating accordingly to their multiplicities, and the corresponding eigenfunctions are denoted, respectively, by \(\{\lambda_{i}\}_{i=1}^{\infty}\) and \(\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}\). Especially, we denote the least positive eigenvalue of it \(\lambda_{1}\) and the normalized positive eigenfunction to \(\lambda_{1}\) \(\varphi_{1}(\Theta)\). In the sequel, for the sake of brevity, we shall write λ and φ instead of \(\lambda_{1}\) and \(\varphi_{1}\), respectively.
The set of sequential eigenfunctions corresponding to the same value of \(\{\lambda_{i}\}_{i=1}^{\infty}\) in the sequence \(\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}\) makes an orthonormal basis for the eigenspace of the eigenvalue \(\lambda_{i}\). Hence for each \(\Omega\subset S^{n-1}\) there is a sequence \(\{k_{j}\}\) of positive integers such that \(k_{1}=1\), \(\lambda_{k_{j}}<\lambda_{k_{j+1}}\), \(\lambda_{k_{j}}=\lambda_{k_{j}+1}=\lambda_{k_{j}+2}=\cdots=\lambda_{k_{j+1}-1}\) and \(\{\varphi_{k_{j}},\varphi_{k_{j}+1},\ldots,\varphi_{k_{j+1}-1}\}\) is an orthonormal basis for the eigenspace of the eigenvalue \(\{\lambda_{k_{j}}\}_{j=1}^{\infty}\). By \(I_{\Omega}(k_{m})\) we denote the set of all positive integers less than \(\{k_{m}\}_{m=1}^{\infty}\). In spite of the fact
the summation over \(I_{\Omega}(k_{1})\) of a function \(S(k)\) of a variable k will be used by promising
If we denote the solutions of the equation
by \(\aleph_{i}^{+}\) and \(\aleph_{i}^{-}\), then the functions
are harmonic functions in \(C_{n}(\Omega)\) and vanish on \(S_{n}(\Omega)\).
Let \(G_{\Omega}(P,Q)\) be the Green function of \(C_{n}(\Omega)\) for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in C_{n}(\Omega)\). Then the Poisson kernel in \(C_{n}(\Omega)\) can be defined by
where \(P\in C_{n}(\Omega)\), \(Q\in S_{n}(\Omega)\), \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into \(C_{n}(\Omega)\) and
Let \(F(\Theta)\) be a function defined in Ω. We denote \(N_{i}(F)\) by
when it exists.
For any two points \(P=(r,\Theta) \) and \(Q=(t,\Phi)\) in \(C_{n}(\Omega)\) and \(S_{n}(\Omega)\), respectively, we define
where m is a non-negative integer and
To obtain the solution of the Dirichlet problem in a cone, as in [1, 3, 4], we use the modified Poisson kernel defined by
where \(P\in C_{n}(\Omega)\) and \(Q\in S_{n}(\Omega)\), which has the following estimates (see [1]):
for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in S_{n}(\Omega)\) satisfying \(0<\frac{r}{t}<\frac{1}{2}\), where M is a constant independent of P, Q, and m. For the construction and applications of a modified Green function in a half space, we refer the reader to the paper by Qiao (see [5]).
Write
where \(f(Q)\) is a continuous function on \(\partial C_{n}(\Omega)\) and \(d\sigma_{Q}\) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(\partial{C_{n}(\Omega)}\).
Recently, Qiao and Deng (cf. [1]) gave the solution of the Dirichlet problem in a cone. Applications of modified Poisson kernel with respect to the Schrödinger operator, we refer the reader to the papers by Huang and Ychussie (see [6]) and Li and Ychussie (see [7]).
Theorem A
If \(\Omega+\aleph^{+}-1>0\), \(\Omega-n+1\leq\aleph_{k_{m+1}}^{+}<\Omega-n+2\) and \(f(Q)\) (\(Q=(t,\Phi )\)) is a continuous function on \(\partial{C_{n}(\Omega)}\) satisfying
then the function \(U_{\Omega}^{m}[f](P)\) is a solution of the Dirichlet problem in \(C_{n}(\Omega)\) with f and
Furthermore, Qiao and Deng (cf. [4]) supplemented the above result and proved the following.
Theorem B
Let \(0< p<\infty\), \(\gamma>(-\aleph^{+}-n+2)p+n-1\) and
If \(f(Q)\) (\(Q=(t,\Phi)\)) is a continuous function on \(S_{n}(\Omega)\) satisfying
then the function \(U_{\Omega}^{m}[f](P)\) satisfies
It is natural to ask if the continuous function u satisfying (2) and (3) can be replaced by arbitrary continuous function? In this paper, we shall give an affirmative answer to this question. To do this, we first construct a modified Poisson kernel. Let \(\phi(l)\) be a positive function of \(l\geq1\) satisfying
Denote the set
by \(\pi_{\Omega}(\phi,i)\). Then \(1\in\pi_{\Omega}(\phi,i)\). When there is an integer N such that \(\pi_{\Omega}(\phi,N)\neq\Phi\) and \(\pi_{\Omega}(\phi,N+1)= \Phi \), denote
of integers. Otherwise, denote the set of all positive integers by \(J_{\Omega}(\phi)\). Let \(l(i)=l_{\Omega}(\phi,i+1)\) be the minimum elements l in \(\pi _{\Omega}(\phi,i)\) for each \(i\in J_{\Omega}(\phi)\). In the former case, we put \(l{(N+1)}=\infty\). Then \(l(1)=1\). The kernel function \(\widetilde{K}_{\Omega}^{\phi}(P,Q)\) is defined by
where \(P\in C_{n}(\Omega)\) and \(Q=(t,\Phi)\in S_{n}(\Omega)\).
The generalized Poisson kernel \(P_{\Omega}^{\phi}(P,Q)\) is defined by
where \(P\in C_{n}(\Omega)\) and \(Q\in S_{n}(\Omega)\).
As an application of the inequality (1) and the generalized Poisson kernel \(PI_{\Omega}^{\phi}(P,Q)\), we have the following.
Theorem
Let \(g(Q)\) be a continuous function on \(S_{n}(\Omega)\). Then there is a positive continuous function \(\phi_{g}(l)\) of \(l\geq0\) depending on g such that
is a solution of the Dirichlet problem in \(C_{n}(\Omega)\) with g.
2 Lemmas
Lemma 1
Let \(\phi(l)\) be a positive continuous function of \(l\geq1\) satisfying
Then
for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in S_{n}(\Omega)\) satisfying
Proof
We can choose two points \(P=(r,\Theta)\in C_{n}(\Omega)\) and \(Q=(t,\Phi)\in S_{n}(\Omega)\), which satisfies (4). Moreover, we also can choose an integer \(i=i(P,Q)\in J_{\Omega}(\phi)\) such that
Then
Hence we have from (1), (4), and (5) that
which is the conclusion. □
Lemma 2
(See [4])
Let \(g(Q)\) be a continuous function on \(\partial{C_{n}(\Omega)}\) and \(V(P,Q)\) be a locally integrable function on \(\partial{C_{n}(\Omega)}\) for any fixed \(P\in C_{n}(\Omega)\), where \(Q\in \partial{C_{n}(\Omega)}\). Define
for any \(P\in C_{n}(\Omega)\) and any \(Q\in\partial{C_{n}(\Omega)}\).
Suppose that the following two conditions are satisfied:
(I) For any \(Q'\in\partial{C_{n}(\Omega)}\) and any \(\epsilon>0\), there exists a neighborhood \(B(Q')\) of \(Q'\) such that
for any \(P=(r,\Theta)\in C_{n}(\Omega)\cap B(Q')\), where R is a positive real number.
(II) For any \(Q'\in\partial{C_{n}(\Omega)}\), we have
for any positive real number R.
Then
for any \(Q'\in\partial{C_{n}(\Omega)}\).
3 Proof of Theorem
Take a positive continuous function \(\phi(l)\) (\(l\geq1\)) such that
and
for \(l>1\), where
For any fixed \(P=(r,\Theta)\in C_{n}(\Omega)\), we can choose a number R satisfying \(R>\max\{1,4r\}\). Then we see from Lemma 1 that
Obviously, we have
which gives
To see that \(H_{\Omega}^{\phi_{g}}(P)\) is a harmonic function in \(C_{n}(\Omega)\), we remark that \(H_{\Omega}^{\phi_{g}}(P)\) satisfies the locally mean-valued property by Fubini’s theorem.
Finally we shall show that
for any \(Q'=(t',\Phi')\in \partial{C_{n}(\Omega)}\). Set
in Lemma 2, which is locally integrable on \(S_{n}(\Omega)\) for any fixed \(P\in C_{n}(\Omega)\). Then we apply Lemma 2 to \(g(Q)\) and \(-g(Q)\).
For any \(\epsilon>0\) and a positive number δ, by (9) we can choose a number R (\(>\max\{1,2(t'+\delta)\}\)) such that (6) holds, where \(P\in C_{n}(\Omega)\cap B(Q',\delta)\).
Since
as \(P=(r,\Theta)\rightarrow Q'=(t',\Phi')\in S_{n}(\Omega)\), we have
where \(Q\in S_{n}(\Omega)\) and \(Q'\in S_{n}(\Omega)\). Then (7) holds.
Thus we complete the proof of Theorem.
References
Qiao, L, Deng, GT: Growth property and integral representation of harmonic functions in a cone. Bull. Malays. Math. Soc. 36(2), 511-523 (2013)
Rosenblum, G, Solomyak, M, Shubin, M: Spectral Theory of Differential Operators. VINITI, Moscow (1989)
Qiao, L: Integral representations for harmonic functions of infinite order in a cone. Results Math. 61(1-2), 63-74 (2012)
Qiao, L: Growth of certain harmonic functions in an n-dimensional cone. Front. Math. China 8(4), 891-905 (2013)
Qiao, L: Modified Poisson integral and Green potential on a half-space. Abstr. Appl. Anal., 2012, Article ID 765965 (2012)
Huang, J, Ychussie, B: The modification of Poisson-Sch integral on cones and its applications. Filomat (to appear)
Li, Z, Ychussie, B: Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator equations. Fixed Point Theory Appl. 2015, 89 (2015)
Acknowledgements
The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions.
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Xue, G., Wang, J. An application of the inequality for modified Poisson kernel. J Inequal Appl 2016, 24 (2016). https://doi.org/10.1186/s13660-016-0959-6
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DOI: https://doi.org/10.1186/s13660-016-0959-6