harmonic

Definitions and Examples

A twice continuously differentiable, complex‑valued function \(u\) defined on \(\Omega\) is \(\textit{harmonic}\) on \(\Omega\) if

\[\Delta u \equiv 0, \]

The simplest nonconstant harmonic functions are the coordinate functions; for example, \(u(x)=x_1\). A slightly more complex example is the function on \(\mathbf{R}^3\) defined by

\[u(x)=x_1^2+x_2^2-2x_3^2+ix_2. \]

Invariance Properties

Throughout this book, all functions are assumed to be complex valued unless stated otherwise. For \(k\) a positive integer, let \(C^k(\Omega)\) denote the set of \(k\) times continuously differentiable functions on \(\Omega\);
\(C^\infty(\Omega)\) is the set of functions that belong to \(C^k(\Omega)\) for every \(k\). For \(E\subset\mathbf{R}^n\), we let \(C(E)\) denote the set of continuous functions on \(E\).

连续k阶倒函数.

For \(y\in\mathbf{R}^n\) and \(u\) a function on \(\Omega\), the \(y\)-translate of \(u\) is the function on \(\Omega+y\) whose value at \(x\) is \(u(x-y)\). Clearly, translations of harmonic functions are harmonic.

平移还是hamonic函数

For a positive number \(r\) and \(u\) a function on \(\Omega\), the \(r\)-dilate of \(u\), denoted \(u_r\), is the function

\[(u_r)(x)=u(rx) \]

defined for \(x\) in \((1/r)\Omega=\{(1/r)w:w\in\Omega\}\). If \(u\in C^2(\Omega)\), then a simple computation shows that \(\Delta(u_r)=r^2(\Delta u)_r\) on \((1/r)\Omega\). Hence dilates of harmonic functions are harmonic.

拉普拉斯算子与正交变换可交换

We now show that the Laplacian commutes with orthogonal transformations; more precisely, if \(T\) is orthogonal and \(u\in C^2(\Omega)\), then

\[\Delta(u\circ T)=(\Delta u)\circ T \]

on \(T^{-1}(\Omega)\)

The Mean‑Value Property

Many basic properties of harmonic functions follow from Green's identity (which we will need mainly in the special case when \(\Omega\) is a ball):

\[\begin{equation} \int_{\Omega}(u\Delta v - v\Delta u)\,dV =\int_{\partial\Omega}\big(u D_{\mathbf{n}}v - v D_{\mathbf{n}}u\big)\,ds. \tag{1.1} \end{equation} \]

Here \(\Omega\) is a bounded open subset of \(\mathbf{R}^n\) with smooth boundary, and \(u\) and \(v\) are \(C^2\)-functions on a neighborhood of \(\overline{\Omega}\), the closure of \(\Omega\). The measure \(V = V_n\) is Lebesgue volume measure on \(\mathbf{R}^n\), and \(s\) denotes surface‑area measure on \(\partial\Omega\) (see Appendix A for a discussion of integration over balls and spheres). The symbol \(D_{\mathbf{n}}\) denotes differentiation with respect to the outward unit normal \(\mathbf{n}\). Thus for \(\zeta \in \partial\Omega\), \((D_{\mathbf{n}}u)(\zeta) = (\nabla u)(\zeta)\cdot \mathbf{n}(\zeta)\), where \(\nabla u = (D_1u,\dots,D_nu)\) denotes the gradient of \(u\) and

\(\cdot\) denotes the usual Euclidean inner product.

这里定义了:
\(s\)、\(\partial\Omega\)、\(\zeta \in \partial\Omega\)、\(D_{\mathbf{n}}\)

\(B(a,r)=\{x\in\mathbf{R}^n:|x-a|<r\}\) is the open ball centered at \(a\) of radius \(r\); its closure is the closed ball \(\overline{B}(a,r)\); the unit ball \(B(0,1)\) is denoted by \(B\) and its closure by \(\overline{B}\).

这里定义了: \(B(a,r)\)、 \(\overline{B}(a,r)\)、\(B\) 、\(\overline{B}\)

Mean‑Value Property:

If \(u\) is harmonic on \(\overline{B}(a,r)\), then \(u\) equals the average of \(u\) over \(\partial B(a,r)\). More precisely,

\[u(a)=\int_{S} u(a+r\zeta)\,d\sigma(\zeta). \]

Volumne measure:

\[\frac{1}{n V(B)} \int_{\mathbf{R}^n} f\,dV =\int_{0}^{\infty} r^{n-1} \int_{S} f(r\zeta)\,d\sigma(\zeta)\,dr \]

More precisely:

If \(u\) is harmonic on \(\overline{B}(a,r)\), then \(u(a)\) equals the average of \(u\) over \(B(a,r)\). More precisely,

\[u(a)=\frac{1}{V(B(a,r))}\int_{B(a,r)} u\,dV. \]

The Poisson kernel for the ball

The mean‑value property shows that if \(u\) is harmonic on \(\overline{B}\), then

\[u(0)=\int_{S} u(\zeta)\,d\sigma(\zeta). \]

We now show that for every \(x\in B\), \(u(x)\) is a weighted average of \(u\) over \(S\). More precisely, we will show there exists a function \(P\) on \(B\times S\) such that

\[u(x)=\int_{S} u(\zeta)\,P(x,\zeta)\,d\sigma(\zeta) \]

for every \(x\in B\) and every \(u\) harmonic on \(\overline{B}\).

更进一步扩展，不只是在圆心.

这里定义了: \(P\)

最后得出:

\[P(x,\zeta)=\frac{1-|x|^{2}}{|x-\zeta|^{n}}. \]

The function \(P\) derived above is called the \(\textit{Poisson kernel}\) for the ball; it plays a key role in the next section.

\(x\in B\): 单位开球内部点, 不是边界; \(|x|<1\)

\(\zeta\in S\): 单位球面上的边界点, \(|\zeta|=1\)

前面：存在函数 \(P(x,\zeta)\)（泊松核），使得

\[u(x)=\int_S u(\zeta)\,P(x,\zeta)\,d\sigma(\zeta) \]

这个式子是对已经调和的 \(u\) 成立。

也就是说：如果 \(u\) 在 \(\overline{B}\) 上调和连续，那么它内部的值可以用边界值乘泊松核积分得到。

这是表示公式：已知解，用边界把内部写出来，不是证明解存在。

The Dirichlet problem for the ball

We now come to a famous problem in harmonic function theory: given a continuous function \(f\) on \(S\), does there exist a continuous function \(u\) on \(\overline{B}\), with \(u\) harmonic on \(B\), such that \(u=f\) on \(S\)? If so, how do we find \(u\)? This is the \(\textit{Dirichlet problem}\) for the ball. Recall that by the maximum principle, if a solution exists, then it is unique.

注意这里:

设 \(\mathbb R^n\) 空间：

1. \(B\)：单位开球

\[B=\{x\in\mathbb R^n\mid |x|<1\} \]

开球：球内部，不包含球面边界。\(u\) harmonic on \(B\)：\(u\) 在球内部调和（\(\Delta u=0\)）。

2. \(S\)：单位球面（\(B\) 的边界）

\[S=\{x\in\mathbb R^n\mid |x|=1\} \]

就是球的外壳曲面；\(f\) 定义在球面 \(S\) 上，是边界条件。

3. \(\overline{B}\)：单位闭球（overline = closure 闭包）

\[\overline{B}=B\cup S=\{x\in\mathbb R^n\mid |x|\le 1\} \]

开球连同它的球面边界一起。

For arbitrary \(f\in C(S)\), we define the Poisson integral of \(f\), denoted \(P[f]\), to be the function on \(B\) given by

\[P[f](x)=\int_S f(\zeta)\,P(x,\zeta)\,d\sigma(\zeta). \]

这里定义了: \(P[f]\)

Solution of the Dirichlet problem for the ball:

Suppose \(f\) is continuous on \(S\). Define \(u\) on \(\overline{B}\) by

\[u(x)= \begin{cases} P[f](x) & \text{if } x\in B\\ f(x) & \text{if } x\in S. \end{cases} \]

We now show that every harmonic function is infinitely differentiable. In dealing with differentiation in several variables the following notation is useful: a multi‑index \(\alpha\) is an \(n\)-tuple of nonnegative integers \((\alpha_1,\dots,\alpha_n)\); the partial differentiation operator \(D^\alpha\) is defined to be \(D_1^{\alpha_1}\dots D_n^{\alpha_n}\) (\(D_j^0\) denotes the identity operator). For each \(\zeta\in S\), the function \(P(\cdot,\zeta)\) is infinitely differentiable on \(B\); we denote its \(\alpha^\text{th}\) partial derivative by \(D^\alpha P(\cdot,\zeta)\) (here \(\zeta\) is held fixed).

这里定义了: multi‑index

Then \(u\) is continuous on \(\overline{B}\) and harmonic on \(B\).

Real Analyticity and Homogeneous Expansions

We saw in the section before last that harmonic functions are infinitely differentiable. A much stronger property will be established in this section—harmonic functions are real analytic. Roughly speaking, a function is real analytic if it is locally expressible as a power series in the coordinate variables \(x_1,x_2,\dots,x_n\) of \(\mathbf R^n\).

To make this more precise, we need to discuss what is meant by a series of complex numbers of the form \(\sum c_\alpha\), where the summation is over all multi‑indices \(\alpha\). (The full range of multi‑indices will always be intended in a series unless indicated otherwise.) The problem is that there is no natural ordering of the set of all multi‑indices when \(n>1\). However, suppose we know that \(\sum c_\alpha\) is absolutely convergent, i.e., that

\[\sup_{F}\sum_{\alpha\in F}|c_\alpha|<\infty, \]

The following notation will be convenient when dealing with multiple power series: for \(x\in\mathbf R^n\) and \(\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)\) a multi‑index, define

\[x^\alpha = x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}, \]

\[\alpha! = \alpha_1!\,\alpha_2!\cdots\alpha_n!, \]

\[|\alpha| = \alpha_1+\alpha_2+\cdots+\alpha_n. \]

A function \(f\) on \(\Omega\) is real analytic on \(\Omega\) if for every \(a\in\Omega\) there exist complex numbers \(c_\alpha\) such that

\[f(x)=\sum c_\alpha (x-a)^\alpha \]

for all \(x\) in a neighborhood of \(a\), the series converging absolutely in this neighborhood.

Theorem:

If \(u\) is harmonic on \(\Omega\), then \(u\) is real analytic in \(\Omega\).

定义: \(q_\alpha\) is a polynomial.

A polynomial is by definition a finite linear combination of monomials \(x^\alpha\). A polynomial \(p\) of the form

\[p(x)=\sum_{|\alpha|=m} c_\alpha x^\alpha \]

is said to be homogeneous of degree \(m\); here we allow \(m\) to be any nonnegative integer. Equivalently, a polynomial \(p\) is homogeneous of degree \(m\) if

\[p(tx)=t^m p(x) \]

for all \(t\in\mathbf R\) and all \(x\in\mathbf R^n\). This last formulation shows that a homogeneous polynomial is determined by its restriction to \(S\): if \(p\) and \(q\) are homogeneous of degree \(m\) and \(p=q\) on \(S\), then \(p=q\) on \(\mathbf R^n\). (This is not true of polynomials in general; for example, \(1-|x|^2\equiv 0\) on \(S\).) Note also that if \(p\) is a homogeneous polynomial of degree \(m\), then so is \(p\circ T\) for every linear map \(T\) from \(\mathbf R^n\) to \(\mathbf R^n\).

Theorem:

Suppose \(u\) is harmonic on \(\Omega\) and \(a\in\Omega\). Then there exist harmonic homogeneous polynomials \(p_m\) of degree \(m\) such that

1.32

\[u(x)=\sum_{m=0}^{\infty} p_m(x-a) \]

for all \(x\) near \(a\), the series converging absolutely and uniformly near \(a\).