On Existence of Boundary Values of Polyharmonic Functions

TL;DR

This paper characterizes the boundary behavior of polyharmonic functions inside the unit disk, establishing conditions for their boundary values to exist in hyperfunction spaces.

Contribution

It provides a comprehensive description of boundary values of polyharmonic functions in terms of hyperfunctions and specifies conditions for these boundary values to belong to particular subspaces.

Findings

Boundary values exist in the space of hyperfunctions.

Necessary and sufficient conditions for boundary value belonging to subspaces.

Complete description of polyharmonic functions inside the unit disk.

Abstract

In trigonometric series terms all polyharmonic functions inside the unit disk are described. For such functions it is proved the existence of their boundary values on the unit circle in the space of hyperfunctions. The necessary and sufficient conditions are presented for the boundary value to belong to certain subspaces of the space of hyperfunctions.

Equations

$${\mathfrak{A}}_{\alpha}=\{\varphi\in D\bigl{|}\exists c>0\ \forall{\mathbb{N}}_{0}\ \max\limits_{t\in\partial K}|{\varphi}^{(k)}(t)|\leq c{\alpha}^{k}k!\}.$$

$$\Vert \phi {\Vert }_{{\mathfrak{A}}_{\alpha }}=\underset{k\in {\mathbb{N}}_0}{\sup }\frac{\underset{t\in \partial K}{\max }|{\phi }^{(k)}(t)|}{{\alpha }^{k}k!}.$$

$$\mathfrak{A}=\underset{\alpha \to \infty }{ind\; lim}{\mathfrak{A}}_{\alpha },$$

$$\mathfrak{A}\subset D\subset L_p(\partial K)=L_p,1\le p<\infty ,$$

$$\sum _{k=-n}^n{c}_k(F)e^{ikt}\stackrel{{\mathfrak{A}}^{\prime }}{\to }F,$$

$$\begin{array}[]{lcl}F\in D&\Longleftrightarrow&\forall\alpha>0\ \exists c>0\quad|c_{k}(F)|\leq c|k|^{-\alpha};\\ F\in{\mathfrak{A}}&\Longleftrightarrow&\exists\alpha>0\ \exists c>0\quad|c_{k}(F)|\leq ce^{-\alpha|k|};\\ F\in{D^{\prime}}&\Longleftrightarrow&\exists\alpha>0\ \exists c>0\quad|c_{k}(F)|\leq c|k|^{\alpha};\\ F\in{\mathfrak{A}}^{\prime}&\Longleftrightarrow&\forall\alpha>0\ \exists c>0\quad|c_{k}(F)|\leq ce^{\alpha|k|}.\end{array}$$

$${\Delta }^{m}u(r,t)=0,0\le r<1,t\in [0,2\pi ].$$

$$u(r,t)=\sum _{j=1}^m(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{c}_k(F_j)r^{|k|}{e}^{ikt},F_j\in {\mathfrak{A}}^{\prime },$$

$$\forall \alpha >0\exists c_j=c_j(\alpha )>0\forall k\in \mathbb{Z}|c_k(F_j)|\le c_j{e}^{\alpha |k|}.$$

$$u(r,t)=\sum _{k=-\infty }^{\infty }{c}_k(r)e^{ikt},c_k(r)=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}u(r,t)e^{-ikt}dt,$$

$$r^2{c_k}^{\prime \prime }(r)+r{c_k}^{\prime }(r)=k^2{c}_k(r).$$

$$c_k(r)=r^{|k|}{c}_k,c_k\in \mathbb{C}.$$

$$\forall r<1r^{|k|}|c_k|=e^{-\alpha |k|}|c_k|\le c,$$

$$u(r,t)=\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(F_1)e^{ikt},F_1\in {\mathfrak{A}}^{\prime }.$$

$$\Delta u(r,t)=\sum _{j=1}^{m-1}(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{c}_k(E_j)r^{|k|}{e}^{ikt}.$$

$$\Delta (u(r,t)-\tilde{u}(r,t))=0,$$

$$u(r,t)=\tilde{u}(r,t)+\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(F_1)e^{ikt},F_1\in {\mathfrak{A}}^{\prime }.$$

$$\Delta u(r,t)=\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(E_1)e^{ikt},E_1\in {\mathfrak{A}}^{\prime }.$$

$$\Delta\left((r^{2}-1)^{j}\sum\limits_{k=-\infty}^{\infty}r^{|k|}c_{k}(F)e^{ikt}\right)=4j\sum\limits_{k=-\infty}^{\infty}\big{[}(r^{2}-1)^{j-1}(|k|+j)+(j-1)(r^{2}-1)^{j-2}\big{]}r^{|k|}c_{k}(F)e^{ikt}$$

$$u_2(r,t)=\frac{1}{4}(r^2-1)\sum _{k=-\infty }^{\infty }\frac{r^{|k|}}{|k|+1}{c}_k(E_1)e^{ikt}$$

$$u(r,t)=\sum _{j=1}^2(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(F_j)e^{ikt}.$$

$$\Delta u(r,t)=\sum _{j=1}^l(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(E_j)e^{ikt},$$

$$\Delta u(r,t)=\sum _{j=1}^{s+1}(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(E_j)e^{ikt}.$$

$$u_{s+2}(r,t)=(r^2-1{)}^{s+1}\sum _{k=-\infty }^{\infty }\frac{r^{|k|}}{4(s+1)(|k|+s+1)}{c}_k(E_{s+1})e^{ikt}.$$

$$\Delta (u-u_{s+2})(r,t)=\sum _{j=1}^{s+1}(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(E_j)e^{ikt}-$$

$$\sum _{k=-\infty }^{\infty }(r^2-1{)}^s{r}^{|k|}{c}_k(E_{s+1})e^{ikt}-\sum _{k=-\infty }^{\infty }\left[\frac{s(r^2-1{)}^{s-1}}{|k|+s+1}+(r^2-1{)}^s\right]r^{|k|}{c}_k(E_{s+1})e^{ikt}=$$

$$\sum _{j=1}^{s-1}(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(E_j)e^{ikt}+(r^2-1{)}^{s-1}\sum _{k=-\infty }^{\infty }{r}^{|k|}\left[c_k(E_s)-\frac{s\cdot c_k(E_{s+1})}{|k|+s+1}\right]e^{ikt}.$$

$$c_k(E_s^{\prime })=c_k(E_s)-\frac{s\cdot c_k(E_{s+1})}{|k|+s+1},$$

$$\Delta (u-u_{s+2})(r,t)=\sum _{j=1}^s(r^2-1{)}^{j-1}\sum _{k=-\infty }^{\infty }{r}^{|k|}{c}_k(E_j^{\prime })e^{ikt},E_j^{\prime }\in {\mathfrak{A}}^{\prime },$$

$$\Delta (u-u_{s+2}-{\tilde{u}}_{s+1})(r,t)=0.$$

Full text

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Moreover, the series $\sum\limits_{k=-\infty}^{\infty}c_{k}(F)e^{ikt}$ converges to $F$ in the corresponding space. The sequence $\{F_{n}\}_{n\in\mathbb{N}}$, whose elements $F_{n}$ belong to one of the spaces $D,{\mathfrak{A}},D^{\prime}$ or ${\mathfrak{A}}^{\prime}$, converges to $F$ in this space if and only if the constants $c$ and $\alpha$ in the above estimates for $|c_{k}(F_{n})|$ do not depend on $n$ and for any $k\in{\mathbb{Z}},\ c_{k}(F_{n})\to c_{k}(F),\ n\to\infty$.

2.  A function $u(r,t)=u(re^{it})\in C^{2m}(K)$ is called $m$-harmonic in $K$ if it satisfies the equation

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Note, that no conditions on the behaviour of $u(r,t)$ near $\partial K$ are imposed.

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be admissible, where $F_{j}$ are uniquely determined by $u(r,t)$.

Proof.  By Proposition 1,

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So the series $\sum\limits_{k=-\infty}^{\infty}c_{k}(F_{j})r^{|k|}e^{ikt}$ converges uniformly in the disk $\overline{K_{R}}=\{z\in{\mathbb{C}}:|z|\leq R\}$ of radius $R<e^{-\alpha}$ and determines an infinitely differentiable function there. The direct check shows that the functions $(r^{2}-1)^{j-1}\sum\limits_{k=-\infty}^{\infty}c_{k}(F_{j})r^{|k|}e^{ikt},\ j=1,2,\dots,m,$  satisfy (1) in $K_{R}$. Since $\alpha>0$ is arbitrary, these functions are solutions of the equation (1) inside $K$.

To prove the necessity, suppose at first $m=1$. Let $u(r,t)$ be a harmonic function in $K$. Then for a fixed $r<1,\ u(r,t)$ is infinitely differentiable in $t$, and it may be written in the form

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where the series and all its derivatives converge uniformly in $t\in[0,2\pi]$. The coefficients $c_{k}(r)$ are infinitely differentiable on [0, 1) and satisfy the equation

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Hence,

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It follows from the convergence of the series in (3) that

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where $\alpha=-\ln r>0$ is arbitrary. By Proposition 1, $c_{k}$ are the Fourier coefficients of a certain hyperfunction $F_{1}$, and

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Thus, the representation (2) is valid when $m=1$.

Assume the representation (2) to be true for an $(m-1)$-harmonic inside $K$ function $u(r,t)\ (m\geq 2)$, and we shall prove that such a representation holds for an $m$-harmonic function.

If $u(r,t)$ is an $m$-harmonic function, then $\Delta u(r,t)$ is an $(m-1)$-harmonic one. By assumption, there exist $E_{j}\in{\mathfrak{A}}^{\prime},\ j=1,2,\dots,m-1,$ such that

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If we choose $\widetilde{u}\in C^{2}(K)$ so that

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then, because of (4), we shall have

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Let us find at first $\widetilde{u}(r,t)$ in the case where the equation (5) is of the form

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By using the identity

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for $F\in{\mathfrak{A}}^{\prime}$, one can verify that the function ${\widetilde{u}}_{2}(r,t)=u_{2}(r,t)$, where

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$({\widetilde{u}}_{1}(r,t)\equiv 0)$, satisfies (6). Set $c_{k}(F_{2})=\frac{c_{k}(E_{1})}{4(|k|+1)}$. By Proposition 1, $F_{2}\in{\mathfrak{A}}^{\prime}$. So, in the case under consideration

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Suppose now that we know solutions $u_{l}(r,t)$ of the equations

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for all $l\leq s,\ s\leq m-2$ is fixed. We show how to find a solution of the equation

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We put

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It follows from (7) and (8) that if $u(r,t)$ is a solution of (9), then

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Taking into account that $E_{s},E_{s+1}\in{\mathfrak{A}}^{\prime}$, we conclude, by Proposition 1, that there exists $E^{\prime}_{s}\in{\mathfrak{A}}^{\prime}$ such that

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whence

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where $E^{\prime}_{j}=E_{j}$ as $j=1,\dots,s-1$. By assumption, we can find ${\widetilde{u}}_{s+1}(r,t)$ so that

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Setting

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we arrive at the equality

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It is not hard to observe that for the desired function $\widetilde{u}(r,t)$ we have the formula

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Then

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where

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Since for $F\in{\mathfrak{A}}^{\prime}$

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we have, by Proposition 1, that

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as $r\to 1$. The elements $F_{j}\in{\mathfrak{A}}^{\prime}$ are determined uniquely by the function $u(r,t)$ in the following way:

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where the limit is taken in the space ${\mathfrak{A}}^{\prime}$. This completes the proof.

Because of harmonicity in $K$ of the functions

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the representation (2) implies, in particular, the next assertion (cf. [4]).

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where the functions $u_{j}(r,t)$ are harmonic in $K$.

When proving the theorem, it was also established the following fact.

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3.   Let $\Phi$ be a complete linear Hausdorff space such that the continuous embeddings

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hold. We say that $F\in\Phi$ is a boundary value on $\partial K$ of an $m$-harmonic in $K$ function $u(r,t)$ and write $F=u(1,\cdot)$ if $u(r,\cdot)\stackrel{{\scriptstyle\Phi}}{{\to}}F$ as $r\to 1$.

It is seen from Theorem 1 and Corollary 2 that every $m$-harmonic in $K$ function has a boundary value in ${\mathfrak{A}}^{\prime}$. Moreover, each element $F\in{\mathfrak{A}}^{\prime}$ is the boundary value of a certain $m$-harmonic in $K$ function. The natural question arises: under what conditions on an $m$-harmonic in $K$ function $u(r,t)$ its boundary value $u(1,\cdot)$ belongs to $\Phi$?

Proof. Necessity.  It is known that if $r_{0}<1$, then $u(r_{0},\cdot)\in\mathfrak{A}$, and $u(r,\cdot)\stackrel{{\scriptstyle\mathfrak{A}}}{{\to}}u(r_{0},\cdot)$ as $1>r\to r_{0}$. Since the embedding ${\mathfrak{A}}\subset\Phi$ is continuous, $u(r,\cdot)\stackrel{{\scriptstyle\Phi}}{{\to}}u(r_{0},\cdot)\ (r\to r_{0})$. By assumption, $u(r,\cdot)\stackrel{{\scriptstyle\Phi}}{{\to}}u(1,\cdot)$ if $r\to 1$. So, the set $\{u(r,\cdot)\}_{r<1}$ is compact in $\Phi$.

Sufficiency.  Let the set $\{u(r,\cdot)\}_{r<1}$ be compact in $\Phi$. Suppose $r\to 1$. Then there exists a subsequence $r_{k}\to 1$ such that $u(r_{k},\cdot)$ converges in $\Phi\ (r_{k}\to 1)$ to a certain element $F\in\Phi$. Since $\Phi\subset{\mathfrak{A}}$ continuously, $u(r_{k},\cdot)$ converges in ${\mathfrak{A}}^{\prime}$. Taking into account that $u(r,\cdot)\stackrel{{\scriptstyle{\mathfrak{A}}^{\prime}}}{{\to}}u(1,\cdot)$ as $r\to 1$, we have $u(1,\cdot)=F\in\Phi$ which completes the proof.

In the partial case where $\Phi=L_{2}(\partial K)$, Theorem 2 was obtained in [7]. By using compactness criteria for sets, one can find the sufficient conditions for the boundary value of a polyharmonic function to belong to $L_{p}(\partial K),\ 1\leq p<\infty$. For instance, the following assertion is valid.

1)  $\sup\limits_{0\leq r<1}\|u(r,\cdot)\|_{L_{p}}<\infty$;

2)  \int\limits_{0}^{2\pi}\big{|}u\big{(}re^{i(t-\tau)}\big{)}-u(re^{it})\big{|}^{p}\,dt\to 0\ (\tau\to 0)\mbox{\rm\ uniformly in}\ r\in[0,1).

Now we consider in more detail the case of $L_{2}$. Let

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The set ${\mathfrak{B}}_{j}$ with norm $\|\cdot\|_{{\mathfrak{B}}_{j}}$ forms a Banach space.

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where $F_{j}$ are taken from representation (2). Moreover, $u(r,\cdot)\to F_{1}\ (r\to 1)$ weakly in the space $L_{2}$.

Proof.  Assume that in the representation (2) $F_{1}\in L_{2},\ F_{j}\in{\mathfrak{B}}_{j-1}\ (j=2,\dots,m)$. Then

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Conversely, let $\sup\limits_{0\leq r<1}\|u(r,\cdot)\|_{L_{2}}<\infty$. Then, as was shown in [4, Lemma 7], each summand in (10) is bounded, too:

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This is equivalent to the inequality

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that is, $F_{1}\in L_{2},\ F_{j}\in{\mathfrak{B}}_{j-1}\ (j=2,\dots,m)$.

It still remains to prove the weak convergence of $u(r,\cdot)$ to $F_{1}\ (r\to 1)$ in $L_{2}$. Since $u(r,\cdot)\stackrel{{\scriptstyle{\mathfrak{A}}^{\prime}}}{{\to}}F_{1}$ as $r\to 1$ and $e^{ikt}\in{\mathfrak{A}}\ (k\in\mathbb{Z})$, we have

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Thus, $u(r,\cdot)\to F_{1}\ (r\to 1)$ weakly in $L_{2}$ on a total set, and $\sup\limits_{0\leq r<1}\|u(r,\cdot)\|_{L_{2}}<\infty$. It follows from here that $u(r,\cdot)\to F_{1}\ (r\to 1)$ weakly in $L_{2}$. The proof is complete.

Let $u(r,t)$ be a harmonic in $K$ function. It follows from (2) that

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In view of $\|u(r,\cdot)\|_{L_{2}}\leq c$, the well-known Fatou lemma and the Lebesgue theorem on passage to the limit yield

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Therefore the weak convergence of $u(r,t)$ to $u(1,t)$ implies the strong one. As was shown in [7], in the case of $m=2$ the boundedness of $\|u(r,\cdot)\|_{L_{2}}$ does not guarantee the convergence of $u(r,\cdot)\ (r\to 1)$ in $L_{2}$.

We pass now to the Sobolev spaces

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The following statement is valid.

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hold.

Proof.  Since the function $f(r)=(1-r^{2})^{2j}r^{2k},\ j,k\in{\mathbb{N}}_{0},$ reaches its maximum at the point $r^{2}=\frac{k}{k+2j}$, and

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we have

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that is, $W_{2}^{-j}\subset{\mathfrak{B}}_{j}$.

Suppose now $F\in{\mathfrak{B}}_{j}$. Then, substituting $z:=r^{2}$,

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Multiplying this inequality by $(1-z)^{-\alpha}$ and then integrating along $[0,1)$, we obtain

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If we put $\delta=2j-\alpha+1$, we get for $n\in\mathbb{N}$

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Since

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the relation $a_{n}=O\left(\frac{1}{n^{\delta}}\right)$ is fulfilled. Taking in (11) $\alpha=1-\varepsilon,\ \varepsilon\in(0,1)$, we conclude that

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that is, $F\in W_{2}^{j-\varepsilon}$, which completes the proof.

The next theorem is devoted to the question on the existence of boundary values in the space $D^{\prime}$ of distributions.

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Proof.  Let the inequality (12) hold. Then for $p\in{\mathbb{N}},\ p>\alpha$, the function $v(r,t)=(1-r^{2})^{p}u(r,t)$ is $(m+p)$-harmonic in $K$, and it is not difficult to verify that

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By Theorems 3,4, the function $v(r,t)$ may be represented in the form (2) where $F_{j}\in W_{2}^{-(j+p)}$. Since $D^{\prime}=\bigcup\limits_{\alpha>0}W_{2}^{-\alpha}$, we have $F_{j}\in D^{\prime}$.

The necessity of condition (12) for $m=1$ was proved in [5]. Namely, it was shown there that for a harmonic function of the form

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there exists $\alpha\geq 0$ such that

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If we take $\alpha=\max\limits_{j}{\alpha}_{j}$, where ${\alpha}_{j}$ corresponds to $F_{j}$ from (2), we obtain the estimate (12) for an $m$-harmonic function ($m$ is arbitrary).

For a number $\beta>1$ we put

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The linear space ${\mathfrak{G}}_{\{\beta\}}$ is endowed with the inductive limit topology of the Banach spaces ${\mathfrak{G}}_{\{\beta,\alpha\}}$ of functions $\varphi\in D$ satisfying (13) with a fixed constant $\alpha$. The norm in ${\mathfrak{G}}_{\{\beta,\alpha\}}$ is defined as

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It is evident, that

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where ${\mathfrak{G}}^{\prime}_{\{\beta\}}$ denotes the dual of ${\mathfrak{G}}_{\{\beta\}}$.

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The proof follows the scheme like that in Theorem 5 if to take into account that

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and the series $\sum\limits_{k=-\infty}^{\infty}c_{k}(F)e^{ikt}$ converges to $F$ in ${\mathfrak{G}}^{\prime}_{\{\beta\}}$-topology.