Modulus of continuity and Heinz-Schwarz type inequalities of solutions to biharmonic equations

TL;DR

This paper derives representation formulas for solutions to inhomogeneous biharmonic equations with boundary conditions, and investigates Heinz-Schwarz inequalities and the modulus of continuity of these solutions.

Contribution

It introduces new representation formulas for biharmonic solutions and extends Heinz-Schwarz inequalities to this context, analyzing their continuity properties.

Findings

Established solution representation formulas for biharmonic equations.

Extended Heinz-Schwarz type inequalities to biharmonic solutions.

Analyzed the modulus of continuity of solutions.

Abstract

For positive integers $n\geq2$ and $m\geq1$, suppose that function $f\in\mathcal{C}^{4}(\mathbb{B}^{n},\mathbb{R}^{m})$ satisfying the following: $(1)$ the inhomogeneous biharmonic equation $\Delta(\Delta f)=g$ ($g\in \mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{m})$) in $\mathbb{B}^{n}$, (2) the boundary conditions $f=\varphi_{1}$ $(\varphi_{1}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m}))$ on $\mathbb{S}^{n-1}$ and $\partial f/\partial\mathbf{n}=\varphi_{2}$ ( $\varphi_{2}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$) on $\mathbb{S}^{n-1}$, where $\partial /\partial\mathbf{n}$ stands for the inward normal derivative, $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$ and $\mathbb{S}^{n-1}$ is the unit sphere of $\mathbb{B}^{n}$. First, we establish the representation formula of solutions to the above inhomogeneous biharmonic Dirichlet problem, and then discuss the Heinz-Schwarz type inequalities and the modulus of continuity of the solutions.

Tables

Table 1. Table 1. Values of U ∗ ​ ( r ​ e n ) superscript 𝑈 ∗ 𝑟 subscript 𝑒 𝑛 U^{\ast}(re_{n}) and U ​ ( r ​ e n ) 𝑈 𝑟 subscript 𝑒 𝑛 U(re_{n}) for Theorem 1.1 .

$n$	$U^{\ast }\left(r{e}_n\right)$	$U\left(r{e}_n\right)$
2	$\frac{2}{\pi }\left(\frac{2r\left(1-r^2\right)}{1+r^2}+\left(1+r^2\right)\arcsin \frac{2r}{1+r^2}\right)$	$\frac{4\arctan r}{\pi }$
3	$\frac{\left(\left(1+4r^2+3r^4\right){\left(1+r^2\right)}^{\frac{1}{2}}+r^6-3r^4+3r^2-1\right)}{3r{\left(1+r^2\right)}^{\frac{3}{2}}}$	$\frac{r^2-1+\sqrt{1+r^2}}{r\sqrt{1+r^2}}$
4	$\frac{2}{\pi }\left(\frac{2r\left(1-r^2\right)}{{\left(1+r^2\right)}^2}+\arcsin \frac{2r}{1+r^2}\right)$	$\frac{2r\left(r^2-1\right)+2{\left(1+r^2\right)}^2\arctan r}{\pi r^2\left(1+r^2\right)}$

Equations

$$\{\begin{array}{ll}\Delta (\Delta f)=g& \text{mbox}in{\mathbb{B}}^n,\\ f={\phi }_1& \text{mbox}on{\mathbb{S}}^{n-1},\\ \frac{\partial f}{\partial \mathbf{n}}={\phi }_2& \text{mbox}on{\mathbb{S}}^{n-1},\end{array}$$

$$\frac{\partial f}{\partial \mathbf{n}}=\left(\frac{\partial f_1}{\partial \mathbf{n}},\dots ,\frac{\partial f_m}{\partial \mathbf{n}}\right)$$

$$\{\begin{array}{ll}\Delta (\Delta F(x))=r^2\Delta (\Delta f(rx)),& x\in {\mathbb{B}}^n,\\ F=f_r& \text{mbox}on{\mathbb{S}}^{n-1},\\ \frac{\partial F}{\partial \mathbf{n}}=r\frac{\partial f_r}{\partial \mathbf{n}}& \text{mbox}on{\mathbb{S}}^{n-1},\end{array}$$

$$f_r(\zeta ):=f(r\zeta ),\zeta \in {\mathbb{S}}^{n-1}.$$

$$H_n(x,\zeta )=\frac{1}{2}\frac{(1-|x{|}^2{)}^2}{|x-\zeta {|}^n}$$

$$\displaystyle K_{n}(x,\zeta)=\frac{1}{4}\frac{(1-|x|^{2})^{2}}{|x-\zeta|^{n+2}}\big{(}n(1-|x|^{2})-(n-4)|x-\zeta|^{2}\big{)}.$$

$$[x,y]:=\left|y|x|-x^{\ast }\right|=\left|x|y|-y^{\ast }\right|\text{mbox}andx\otimes y:=(1-|x{|}^2)(1-|y{|}^2).$$

$$G_{2,n}(x,y)=\{\begin{array}{ll}{c}_n\left(|x-y{|}^{4-n}-[x,y{]}^{4-n}-\frac{(n-4)}{2}(x\otimes y)[x,y{]}^{2-n}\right),& n\ne 2,4,\\ & \\ c_n\left(|x-y{|}^{4-n}\log \frac{|x-y{|}^2}{[x,y{]}^2}+(x\otimes y)[x,y{]}^{2-n}\right),& n=2,4,\end{array}$$

$$K[{\phi }_1](x)={\int }_{{\mathbb{S}}^{n-1}}{K}_n(x,\zeta ){\phi }_1(\zeta )d\sigma (\zeta ),H[{\phi }_2](x)={\int }_{{\mathbb{S}}^{n-1}}{H}_n(x,\zeta ){\phi }_2(\zeta )d\sigma (\zeta )$$

$$G[g](x)={\int }_{{\mathbb{B}}^n}{G}_{2,n}(x,y)g(y)dV(y),$$

$$f(x)=K[{\phi }_1](x)+H[{\phi }_2](x)+G[g](x),x\in {\mathbb{B}}^n.$$

$$|f(z)|\le \frac{4}{\pi }\arctan |z|,z\in \mathbb{D}.$$

$$\left|f(z)-\frac{1-|z{|}^2}{1+|z{|}^2}f(0)\right|\le \frac{4}{\pi }\arctan |z|,z\in \mathbb{D},$$

$$|f(x)|\le U(|x|e_n),$$

$$U(x)=P[{\chi }_{{\mathbb{S}}_{+}^{n-1}}-{\chi }_{{\mathbb{S}}_{-}^{n-1}}](x):={\int }_{{\mathbb{S}}^{n-1}}\frac{1-|x{|}^2}{|x-\zeta {|}^n}\left({\chi }_{{\mathbb{S}}_{+}^{n-1}}(\zeta )-{\chi }_{{\mathbb{S}}_{-}^{n-1}}(\zeta )\right)d\sigma (\zeta ).$$

$$\left|f(x)-\frac{1-|x{|}^2}{(1+|x{|}^2{)}^{\frac{n}{2}}}f(0)\right|\le U(|x|e_n),x\in {\mathbb{B}}^n.$$

$$\displaystyle\big{|}f(x)$$

$${\delta }_1(|x|)=\frac{n}{4}\frac{(1-|x{|}^2{)}^3}{(1+|x{|}^2{)}^{\frac{n+2}{2}}}-\frac{(n-4)}{4}\frac{(1-|x{|}^2{)}^2}{(1+|x{|}^2{)}^{\frac{n}{2}}},{\delta }_2(|x|)=\frac{(1-|x{|}^2{)}^2}{(1+|x{|}^2{)}^{\frac{n}{2}}}$$

$$U^{\ast }(x)={\int }_{{\mathbb{S}}^{n-1}}\frac{(1-|x{|}^2{)}^3}{|x-\zeta {|}^{n+2}}\left({\chi }_{{\mathbb{S}}_{+}^{n-1}}(\zeta )-{\chi }_{{\mathbb{S}}_{-}^{n-1}}(\zeta )\right)d\sigma (\zeta ).$$

$$\underset{x,y\in \Omega ,x\ne y}{\sup }\frac{|u(x)-u(y)|}{\omega (|x-y|)}\le L.$$

$$P[\psi ](z)=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{1-|z{|}^2}{|e^{i\theta }-z{|}^2}u(e^{i\theta })d\theta ,z\in \mathbb{D}.$$

$$\underset{t\to 0^{+}}{\mathrm{limsup}}\frac{\omega (t)}{t}=c<\infty .$$

$${}_2{F}_1(a,b;c;t)=\sum _{k=0}^{\infty }\frac{(a{)}_k(b{)}_k}{(c{)}_k}\frac{t^k}{k!}$$

$${\varphi }_x(y)=\frac{|x-y{|}^{2}x-(1-|x{|}^2)(y-x)}{[x,y{]}^2},y\in {\mathbb{B}}^n.$$

$$|{\varphi }_x(y)|=\frac{|x-y|}{[x,y]},1-|{\varphi }_x(y){|}^2=\frac{(1-|x{|}^2)(1-|y{|}^2)}{[x,y{]}^2}$$

$$|J_{{\varphi }_x}(y)|=\frac{(1-|x{|}^2{)}^n}{[x,y{]}^{2n}}.$$

$$|A|=\underset{x\ne 0}{\sup }\frac{|Ax|}{|x|}=\max \{|A\theta |:\theta \in {\mathbb{S}}^{n-1}\},$$

$$l(A)=\inf \{|A\theta |:\theta \in {\mathbb{S}}^{n-1}\}.$$

$$D_{f}=\left(\begin{array}[]{cccc}\displaystyle D_{1}f_{1}\;\cdots\;D_{n}f_{1}\\[11.38109pt] \vdots\;\;\;\;\cdots\;\;\;\;\vdots\\[5.69054pt] \displaystyle D_{1}f_{m}\;\cdots\;D_{n}f_{m}\end{array}\right)=(\nabla f_{1}~{}\cdots,\nabla f_{m})^{T},$$

$${\zeta }_1$$

Full text

∗ Corresponding author (E-mail address:  mathechen@126.com)

1. Introduction and main results

1.1. Notations

For a positive integer $n\geq 2$, let $\mathbb{R}^{n}$ and $\mathbb{R}=\mathbb{R}^{1}$ be the usual real vector space of dimension $n$ and the set of real numbers, respectively. Let $\mathbb{B}^{n}(x_{0},r)=\{x\in\mathbb{R}^{n}:~{}|x-x_{0}|<r\}$, $\overline{\mathbb{B}^{n}}(x_{0},r)=\{x\in\mathbb{R}^{n}:~{}|x-x_{0}|\leq r\}$ and $\mathbb{S}^{n-1}(x_{0},r)=\partial\mathbb{B}^{n}(x_{0},r)$, where $r>0$. We write $\mathbb{B}^{n}:=\mathbb{B}^{n}(0,1)$ and $\mathbb{S}^{n-1}:=\mathbb{S}^{n-1}(0,1)$. Set $\mathbb{D}=\mathbb{B}^{2}$, the open unit disk in the complex plane $\mathbb{C}\cong\mathbb{R}^{2}$. For $m\in\mathbb{N}:=\{1,2,\ldots\}$ and $k\in\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, we denote by $\mathcal{C}^{k}(\Omega_{1},\Omega_{2})$ the set of all $k$-times continuously differentiable functions from $\Omega_{1}$ into $\Omega_{2}$, where $\Omega_{1}$ and $\Omega_{2}$ are subsets of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively. In particular, let $\mathcal{C}(\Omega_{1},\Omega_{2}):=\mathcal{C}^{0}(\Omega_{1},\Omega_{2})$, the set of all continuous functions of $\Omega_{1}$ into $\Omega_{2}$.

1.2. Inhomogeneous biharmonic

equation

For $n\geq 2$ and $m\geq 1$, let $\varphi_{1}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$, $\varphi_{2}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$ and $g\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{m})$. Of particular interest to us is the following inhomogeneous biharmonic problem:

[TABLE]

where $\Delta:=\sum_{j=1}^{n}\frac{\partial^{2}}{\partial x_{j}^{2}}$ is the Laplace operator,

[TABLE]

and $\partial f_{k}/\partial\mathbf{n}$ denotes the differentiation in the inward normal direction for $k\in\{1,\ldots,m\}$. Here the boundary conditions in (1.1) are interpreted in the following distributional sense. For some fixed $r\in(0,1)$, let $F(x)=f(rx),~{}x\in\mathbb{B}^{n}$. Then

[TABLE]

and $f_{r}\rightarrow\varphi_{1}$ as $r\rightarrow 1^{-}$, and r\big{(}\partial f_{r}/\partial\mathbf{n}\big{)}\rightarrow\varphi_{2} as $r\rightarrow 1^{-}$, where

[TABLE]

In particular, if $g\equiv 0$, then the solutions to (1.1) are biharmonic mappings (see [5, 6, 12, 26]).

The inhomogeneous biharmonic equations arise in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows (cf. [13, 21, 27, 28]). This article continues the study of the previous work of Li et al. [22], Kalaj [18] and the monograph of Gazzola et al. [10]. In order to state our main results, we introduce some necessary terminologies.

For $x\in\mathbb{B}^{n}$ and $\zeta\in\mathbb{S}^{n-1}$, let

[TABLE]

and

[TABLE]

Here $K_{n}$ is called a biharmonic Poisson kernel (see [10, p.157]).

For $x,y\in\mathbb{R}^{n}\backslash\{0\}$, we define $x^{\ast}=x/|x|$, $y^{\ast}=y/|y|$,

[TABLE]

Also, for $x,y\in\mathbb{B}^{n}$ with $x\neq y$, we use $G_{2,n}(x,y)$ to denote the biharmonic Green function:

[TABLE]

where A_{n-1}=2\pi^{\frac{n}{2}}/\Gamma\big{(}\frac{n}{2}\big{)} is the $(n-1)$-dimensional surface area of $\mathbb{S}^{n-1}$, c_{n}=1/\big{(}2(4-n)(2-n)A_{n-1}\big{)} for $n\neq 2,4$, and c_{n}=1/\big{(}8(-1)^{n/2+1}A_{n-1}\big{)} for $n=2,4$. We refer readers to the Chapter 4 in [10] for general properties of the Green functions.

1.3. Main results

For $n=2$, Li and Ponnusamy ([22, Theorem 1.1]) established a representation formula and the uniqueness of the solutions to (1.1). In fact, for $n\geq 2$, the solutions to (1.1) with smooth boundary conditions has alreadly been observed in [10, p.138]. For the sake of completeness, we recall the representation formula and the uniqueness of the solutions to (1.1) for some slightly weaker boundary value conditions.

Proposition 1.1**.**

For positive integers $n\geq 2$ and $m\geq 1$, suppose that $\varphi_{1}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$, $\varphi_{2}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$ and $g\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{m})$. Let

[TABLE]

and

[TABLE]

where $d\sigma$ denotes the normalized Lebesgue surface measure on $\mathbb{S}^{n-1}$ and $dV$ is the Lebesgue volume measure on $\mathbb{B}^{n}$. If $f$ is a solution to (1.1), then

[TABLE]

Heinz in his classical paper [14] showed that the following result which is called the Heinz-Schwarz type inequality of harmonic mappings: If $f$ is a harmonic mapping of $\mathbb{D}$ into $\mathbb{D}$ with $f(0)=0$, then

[TABLE]

Later, Hethcote [15] removed the assumption $f(0)=0$ and proved the following inequality

[TABLE]

where $f$ is a harmonic mapping from $\mathbb{D}$ into itself (see also [23, Theorem 3.6.1]). For $n\geq 3$, the classical Heinz-Schwarz type inequality of harmonic mappings in $\mathbb{B}^{n}$ infers that if $f$ is a harmonic mapping of $\mathbb{B}^{n}$ into itself satisfying $f(0)=0,$ then

[TABLE]

where $e_{n}=(0,\ldots,0,1)$ and $U$ is a harmonic function of $\mathbb{B}^{n}$ into $[-1,1]$ defined by

[TABLE]

Here $\chi$ is the indicator function and $\mathbb{S}^{n-1}_{+}=\{x\in\mathbb{S}^{n-1}:~{}x_{n}\geq 0\}$, $\mathbb{S}^{n-1}_{-}=\{x\in\mathbb{S}^{n-1}:~{}x_{n}\leq 0\}$ (cf. [2]). In [18], Kalaj showed the following result for harmonic mappings $f$ of $\mathbb{B}^{n}$ into itself:

[TABLE]

By analogy with the inequality (1.4), we obtain the following result.

Theorem 1.1**.**

For positive integers $n\geq 2$ and $m\geq 1$, let $\varphi_{1}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$, $\varphi_{2}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$ and $g\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{m})$. If $f$ is a solution to (1.1), then, for $x\in\overline{\mathbb{B}^{n}}$,

[TABLE]

where $\|\varphi_{k}\|_{\infty}=\sup_{\zeta\in\mathbb{S}^{n-1}}|\varphi_{k}(\zeta)|~{}(k=1,2),$ $\|g\|_{\infty}=\sup_{x\in\mathbb{B}^{n}}|g(x)|,$

[TABLE]

and

[TABLE]

Moreover, if $g=(8n(2+n)M,0,\ldots,0)\in\mathbb{R}^{m}$ in $\mathbb{B}^{n}$, and $\varphi_{1}=\varphi_{2}=(0,\ldots,0)\in\mathbb{R}^{m}$ in $\mathbb{S}^{n-1}$, then $f(x)=\left(M(1-|x|^{2})^{2},0,\ldots,0\right)\in\mathbb{R}^{m}$ shows that the estimate of (1.1) is sharp in $\overline{\mathbb{B}^{n}}$, where $M$ is a constant.

We remark that Theorem 1.1 is somehow weakened by the fact that the function $\delta_{1}(|x|)$ may change sign; this is in contrast with what happens in the second order case (see (1.4)).

A continuous increasing function $\omega:\,[0,\infty)\rightarrow[0,\infty)$ with $\omega(0)=0$ is called a majorant if $\omega(t)/t$ is non-increasing for $t>0$ (cf. [7, 8]). Given a subset $\Omega$ of $\mathbb{R}^{n}$, a function $u:~{}\Omega\rightarrow\mathbb{R}^{m}$ is said to belong to the Lipschitz space $\mathcal{L}_{\omega}(\Omega,\mathbb{R}^{m})$ if there is a positive constant $L$ such that

[TABLE]

Dyakonov [7, 8] characterized the analytic functions of class $\mathcal{L}_{\omega}(\mathbb{D},\mathbb{C})$ in terms of their modulus (see also [24]).

It is well-known that the condition $u\in\mathcal{L}_{\omega}(\mathbb{S}^{1},\mathbb{C})$ is not enough to guarantee that its harmonic extension $P[\psi]$ belongs to $\mathcal{L}_{\omega}(\mathbb{D},\mathbb{C})$, where $\omega(t)=t$ and

[TABLE]

In fact, $P[\psi]\in\mathcal{L}_{\omega}(\mathbb{D},\mathbb{C})$ is Lipschitz continuous if and only if the Hilbert transform of $d\psi(e^{i\theta})/d\theta$ belongs to $L^{\infty}(\mathbb{S}^{1})$ (see [1] and [29]), where $\omega(t)=t$. In [1], Arsenović et al. established the following result for harmonic mappings of $\mathbb{B}^{n}$ into $\mathbb{R}^{n}$: For a boundary function which is Lipschitz continuous, if its harmonic extension is quasiregular, then this extension is also Lipschitz continuous. Recently, the relationship of the Lipschitz continuity between the boundary functions and their harmonic extensions has attracted much attention (see [4, 5, 16, 19, 22]). Li and Ponnusamy [22] discussed the Lipschitz characteristic of solutions to the inhomogeneous biharmonic equation (1.1) for $n=2$. The same problem in higher dimentional space is much more complicated because of the lack of the techniques of complex analysis. For $n\geq 2$, we will investigate the Lipschitz continuity (or the modulus of continuity) of the solutions to (1.1) as follows.

Theorem 1.2**.**

Suppose that $n\geq 2$ and $m\geq 1$ are integers, and $\omega$ is a majorant satisfying

[TABLE]

For $\varphi_{1}\in\mathcal{L}_{\omega}(\mathbb{S}^{n-1},\mathbb{R}^{m})$, $\varphi_{2}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$ and $g\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{m}),$ if $f$ satisfies (1.1), then $f\in\mathcal{L}_{\omega}(\mathbb{B}^{n},\mathbb{R}^{m}).$

The rest of this article is organized as follows. In section 2, some necessary notations and useful results will be introduced. In section 3, the Proposition 1.1 and Theorem 1.1 will be proved. Theorem 1.2 will be showed in section 4.

2. Preliminaries

2.1. Gauss Hypergeometric Functions

For $a,b,c\in\mathbb{R}$ with $c\neq 0,-1,-2,\ldots,$ the hypergeometric function is defined by the power series

[TABLE]

with respect to the variable $t\in(-1,1)$. Here $(a)_{0}=1$, $(a)_{k}=a(a+1)\cdots(a+k-1)$ for $k=1,2,\ldots$, and generally $(a)_{k}=\Gamma(a+k)/\Gamma(a)$ is the Pochhammer symbol, where $\Gamma$ is the Gamma function (cf. [25]).

2.2. Möbius Transformations of $\mathbb{B}^{n}$

For any fixed $x\in\mathbb{B}^{n}$, the Möbius transformation in $\mathbb{B}^{n}$ is defined by

[TABLE]

The set of isometries of the hyperbolic unit ball is a Kleinian subgroup of all Möbius transformations of the extended spaces $\mathbb{R}^{n}\cup\{\infty\}$ onto itself. In the following, we make use of the automorphism group ${\operatorname{Aut}}(\mathbb{B}^{n})$ consisting of all Möbius transformations of the unit ball $\mathbb{B}^{n}$ onto itself. We recall the following facts from [3]: For $x\in\mathbb{B}^{n}$ and $\phi_{x}\in{\operatorname{Aut}}(\mathbb{B}^{n})$, we have $\phi_{x}(0)=x$, $\phi_{x}(x)=0$, $\phi_{x}(\phi_{x}(y))=y\in\mathbb{B}^{n}$,

[TABLE]

and

[TABLE]

2.3. Matrix notations

For an $m\times n$ matrix $A=(a_{ij})_{m\times n}$, the operator norm of $A$ is defined by

[TABLE]

and the matrix function $l(A)$ is defined by

[TABLE]

For a domain $\Omega\subset\mathbb{R}^{n}$, let $f=(f_{1},\ldots,f_{m}):~{}\Omega\rightarrow\mathbb{R}^{m}$ be a function that has all partial derivatives at $x=(x_{1},\ldots,x_{n})$ in $\Omega$. Then we denote the derivative $D_{f}$ of $f$ by

[TABLE]

where $D_{j}f_{i}(x)=\partial f_{i}(x)/\partial x_{j}$, $T$ is the transpose and the gradients $\nabla f_{j}~{}(j=1,\ldots,m)$ are understood as column vectors.

2.4. Spherical coordinate transformation

Let $Q=(\zeta_{1},\ldots,\zeta_{n}):~{}\mathbb{T}^{n-1}\rightarrow\mathbb{S}^{n-1}$ be the following spherical coordinate transformation

[TABLE]

where $\mathbb{T}^{n-1}=\underbrace{[0,\pi]\times\cdots\times[0,\pi]}\times[0,2\pi].$ We use

[TABLE]

to denote the Jacobian of $Q$.

3. The Heinz-Schwarz type inequalities of solutions to inhomogeneous

biharmonic Dirichlet problems

The proof of Proposition 1.1

For some fixed $r\in(0,1)$, let $F(x)=f(rx),~{}x\in\mathbb{B}^{n}$. It follows from [10, Formula 4.98] that

[TABLE]

is the only solution to (1.2), where $g^{\ast}(x)=r^{2}g(rx),~{}x\in\mathbb{B}^{n}$, $\varphi_{1}^{\ast}(\zeta)=f_{r}(\zeta):=f(r\zeta)$ and $\varphi_{2}^{\ast}(\zeta)=r\frac{\partial f_{r}(\zeta)}{\partial\mathbf{n}},~{}\zeta\in\mathbb{S}^{n-1}$. By letting $r\rightarrow 1^{-}$, we get the desired result. ∎

The following result will be used in the proof of Theorem 1.1.

Lemma A. ([17] or [25, 2.5.16(43)])* For $\mu_{1}>1$ and $\mu_{2}>0$, we have*

[TABLE]

*where $\mathbf{B}(.,.)$ denotes the beta function. *

The proof of Theorem 1.1

For $x\in\mathbb{B}^{n}$, let

[TABLE]

Then $\Phi$ is biharmonic in $\mathbb{B}^{n}$.

We first assume that $x=|x|e_{n}$ is on the ray $[0,e_{n}]$, where $e_{n}=(0,\ldots,0,1)$. Then we have

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

By calculations, we obtain

[TABLE]

and

[TABLE]

It follows from (3.1), (3) and (3) that

[TABLE]

where

[TABLE]

If $x$ is not on the ray $[0,e_{n}]$, then we choose a unitary transformation $\mathcal{O}$ such that $\mathcal{O}(e_{n})=x/|x|$. By making use of the biharmonic mapping $\mathcal{B}(y)=\Phi(\mathcal{O}(y))$, we get

[TABLE]

which, together with (3), implies that, for any $x\in\mathbb{B}^{n}$,

[TABLE]

Next, we estimate $|G[g]|$.

$\mathbf{Case~{}1.}$ $n\neq 2,4$.

For $x,y\in\mathbb{B}^{n}$, let $z=\phi_{x}(y)\in{\operatorname{Aut}}(\mathbb{B}^{n})$. Then

[TABLE]

which, together with (2.2), gives

[TABLE]

By [11, Lemma 2.1], we know that $G_{2,n}(x,y)>0$. Then, by changing variables, we obtain

[TABLE]

where

[TABLE]

and

[TABLE]

Now we estimate $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$. Using the spherical coordinates and Lemma , we obtain

[TABLE]

which gives that

[TABLE]

and

[TABLE]

It follows from (3.7), (3) and (3) that

[TABLE]

$\mathbf{Case~{}2.}$ $n=2$.

For $x,y\in\mathbb{B}^{2}$, let $z=\phi_{x}(y)\in{\operatorname{Aut}}(\mathbb{B}^{2})$. By (2.2), (2.3) and Lemma , we have

[TABLE]

which, together with $G_{2,2}(x,y)>0,$ yields that

[TABLE]

$\mathbf{Case~{}3.}$ $n=4$.

Since $G_{2,4}(x,y)>0~{}\mbox{for}~{}x,y\in\mathbb{B}^{4}$, by (3), we see that

[TABLE]

where $z=\phi_{x}(y)$. Hence (1.1) follows from (3), (3), (3) and (3).

In particular, for $n=2,3,4$, we compute the values of $U^{\ast}(re_{n})$ and $U(re_{n})$, repectively, where $r=|x|$. Let $\zeta=(\zeta_{1},\ldots,\zeta_{n})\in\mathbb{S}^{n-1}$ such that $\zeta_{n}=\cos\theta$, where $\theta$ is the angle between the vector $x$ and $x_{n}$ axis. Let $m(r)=2r/(1+r^{2})$. Elementary calculations lead to

[TABLE]

and

[TABLE]

which imply that

[TABLE]

where

[TABLE]

Since $(-1)^{k}-1=0$ for $k=2j$, where $j\in\{0,1,\ldots,\}$, we see that $Q(r)$ can be rewritten as

[TABLE]

where ${{}_{4}}F_{3}$ is defined in [9]. By applying [9, Eq. 3.1.8] to (3.14), we obtain the values of $U^{\ast}(re_{n})$ (see the Table 1). The values of $U(re_{n})$ follows from [18, Remark 2.7] (see also the Table 1). The proof of this theorem is complete. ∎

4. modulus of continuity of solutions to the inhomogeneous

biharmonic Dirichlet problems

We begin this part with the following two Lemmas which will be used in the proof of Theorem 1.2.

Lemma 4.1**.**

For $x\in\mathbb{B}^{n}$,

[TABLE]

Proof. By the spherical coordinate transformation (see section 2.4) and Lemma , we have

[TABLE]

Elementary computations show that

[TABLE]

which, together with (4), implies that

[TABLE]

The proof of this lemma is finished. ∎

Lemma B. ([20, Lemma 2.5])* Let $\varrho$ be a bounded (absolutely) integrable function defined on a bounded domain $\Omega\subset\mathbb{R}^{n}$. Then the potential type integral*

[TABLE]

belongs to the space $\mathcal{C}^{k}(\mathbb{R}^{n},\mathbb{R})$, where $k+\alpha<n.$ Moreover,

[TABLE]

The proof of Theorem 1.2

We divide the proof of this theorem into four steps.

Step 4.1**.**

The estimate of $|D_{K[\varphi_{1}]}|$.

For $x=(x_{1},\ldots,x_{n})\in\mathbb{B}^{n}$ and $\zeta=(\zeta_{1},\ldots,\zeta_{n})\in\mathbb{S}^{n-1}$, we obtain

[TABLE]

where $k\in\{1,\ldots,n\}$.

Then, for any $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}\setminus\{0\}$, we obtain

[TABLE]

which, together with Cauchy-Schwarz’s inequality, implies that

[TABLE]

where $\langle\cdot,\cdot\rangle$ is the Euclidean inner product.

By (4) and Lemma 4.1, for $x=|x|x^{\ast}\in\mathbb{B}^{n}\setminus\{0\}$, we get

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Now we first estimate $I_{1}$ and $I_{2}$. By (4) and Cauchy-Schwarz’s inequality, we have

[TABLE]

Since $\varphi_{1}\in\mathcal{L}_{\omega}(\mathbb{S}^{n-1},\mathbb{R}^{m})$, we see that there is positive constant $L$ such that

[TABLE]

Simple calculations show that, for $\zeta\in\mathbb{S}^{n-1}$,

[TABLE]

By (4.4), (4.5) and (4.6), we have

[TABLE]

where

[TABLE]

Applying (4), (4.5) and (4.6), we obtain

[TABLE]

where

[TABLE]

Next, we estimate $I_{3}$ and $I_{4}$. By (4.5) and (4.6), we get

[TABLE]

and

[TABLE]

It follows from (4), (4), (4), (4.9) and (4) that

[TABLE]

Step 4.2**.**

The estimate of $|D_{H[\varphi_{2}]}|$.

For $x=(x_{1},\ldots,x_{n})\in\mathbb{B}^{n}$ and $\zeta=(\zeta_{1},\ldots,\zeta_{n})\in\mathbb{S}^{n-1}$, we obtain

[TABLE]

where $k\in\{1,\ldots,n\}$. Then, for any $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}\setminus\{0\}$, we have

[TABLE]

which, together with Cauchy-Schwarz’s inequality, gives that

[TABLE]

Applying (4.4) and (4.12), we see that

[TABLE]

where

[TABLE]

Step 4.3**.**

The estimate of $|D_{G[g]}|$.

$\mathbf{Case~{}1.}$ $n\neq 2,4$.

For $x=(x_{1},\ldots,x_{n}),y=(y_{1},\ldots,y_{n})\in\mathbb{B}^{n}$ with $x\neq y$, we have

[TABLE]

Then, for any $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}\setminus\{0\}$, we get

[TABLE]

which, together with Lemma , yields that

[TABLE]

where

[TABLE]

and

[TABLE]

Now we estimate $I_{5}$. Let $z=\phi_{x}(y)\in{\operatorname{Aut}}(\mathbb{B}^{n})$. Then, by (2.3) and (3.6), we have

[TABLE]

It follows from (4) that

[TABLE]

which, together with (4), implies that

[TABLE]

Next, we estimate $I_{6}$. By (2.2), we have

[TABLE]

which gives that

[TABLE]

Applying (3.6) and (4.18), we see that

[TABLE]

which, together with (3.6) and (4), implies that

[TABLE]

At last, we estimate $I_{7}$ and $I_{8}$. It follows from (4) that

[TABLE]

Since

[TABLE]

by (4), we see that

[TABLE]

By (4.17), (4.20), (4.21) and (4), we conclude that

[TABLE]

$\mathbf{Case~{}2.}$ $n=2$.

For $x=(x_{1},x_{2}),y=(y_{1},y_{2})\in\mathbb{B}^{n}$ with $x\neq y$, we have

[TABLE]

which gives that, for any $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2}\setminus\{0\}$,

[TABLE]

where $k\in\{1,2\}$.

Then, by (4) and Lemma , we have

[TABLE]

where

[TABLE]

Now we estimate $I_{9}$. By Cauchy-Schwarz’s inequality and Lemma , we see that

[TABLE]

Let $z=\phi_{x}(y)\in{\operatorname{Aut}}(\mathbb{B}^{n})$. Then, by (3.6) and (4), we get

[TABLE]

At last, we estimate $I_{10}$. Since $\phi_{x}(y)\in{\operatorname{Aut}}(\mathbb{B}^{n})$, we see that

[TABLE]

Hence, in this case, it follows from (4), (4) and (4.28) that there is a positive constant $M_{4}$ such that

[TABLE]

$\mathbf{Case~{}3.}$ $n=4$.

For $x=(x_{1},x_{2},x_{3},x_{4}),y=(y_{1},y_{2},y_{3},y_{4})\in\mathbb{B}^{n}$ with $x\neq y$, we have

[TABLE]

which yields that, for any $\xi=(\xi_{1},\xi_{2},\xi_{3},\xi_{4})\in\mathbb{R}^{n}\setminus\{0\}$,

[TABLE]

where $k\in\{1,2,3,4\}$.

Next, we estimate

[TABLE]

and

[TABLE]

First, we know from elementary calculations that

[TABLE]

Let $z=\phi_{x}(y)\in{\operatorname{Aut}}(\mathbb{B}^{4})$. Then, by (2.2), (2.3), (3.6) and (4), we obtain

[TABLE]

Since

[TABLE]

we see that

[TABLE]

It follows from (4), (4), (4), (4.33) and Lemma that

[TABLE]

Therefore, by (4.23), (4.29) and (4), we conclude that there exists a positive constant $M_{5}$ such that

[TABLE]

Step 4.4**.**

The Lipschitz continuity of $f$.

By (4.11), (4) and (4.35), we see that there is a constant $M_{6}(n,\|\varphi_{2}\|_{\infty},\|g\|_{\infty})$ such that

[TABLE]

which yields that, for any $a,b\in\mathbb{B}^{n}$,

[TABLE]

The proof of this theorem is complete. ∎

Acknowledgements: This research was partly supported by the Hunan Provincial Education Department Outstanding Youth Project (No. 18B365), the Science and Technology Plan Project of Hengyang City (No. 2018KJ125), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020), the Science and Technology Plan Project of Hengyang City (No. 2017KJ183), and the Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469).