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| Some real functions in~$\mathbb{R}^n$ (e.g. any linear function, or any affine function) are obviously harmonic functions. What are some more interesting harmonic functions? |
Some real functions in~$\mathbb{R}^n$ (e.g. any linear function, or any affine function) are obviously harmonic functions. What are some more interesting harmonic functions? |
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| \begin{itemize} |
\begin{itemize} |
| \item |
\item |
| For~$n\ge 3$, define (on the punctured space~$U=\mathbb{R}^n \setminus \{0\}$) the function~$f(x)=\norm{x}^{2-n}$. Then |
For~$n\ge 3$, define (on the punctured space~$U=\mathbb{R}^n \setminus \{0\}$) the function~$f(x)=\norm{x}^{2-n}$. Then |
| $$ |
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| \frac{\partial f}{\partial x_i} = (2-n) \frac{x_i}{\norm{x}^n}, |
\frac{\partial f}{\partial x_i} = (2-n) \frac{x_i}{\norm{x}^n}, |
| $$ |
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| and |
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| $$ |
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| \frac{\partial^2 f}{{\partial x_i}^2} = |
\frac{\partial^2 f}{{\partial x_i}^2} = |
| n(n-2)\frac{x_i^2}{\norm{x}^{n+2}} - (n-2)\frac{1}{\norm{x}^n} |
n(n-2)\frac{x_i^2}{\norm{x}^{n+2}} - (n-2)\frac{1}{\norm{x}^n} |
| $$ |
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| Summing over $i=1,...,n$ shows $\Delta f\equiv 0$. |
Summing over $i=1,...,n$ shows $\Delta f\equiv 0$. |
| \item |
\item |
| For~$n=2$, define (on the punctured plane~$U=\mathbb{R}^2 \setminus \{0\}$) the function~$f(x,y)=\log(x^2+y^2)$. Derivation and summing yield~$\Delta f\equiv 0$. |
For~$n=2$, define (on the punctured plane~$U=\mathbb{R}^2 \setminus \{0\}$) the function~$f(x,y)=\log(x^2+y^2)$. Derivation and summing yield~$\Delta f\equiv 0$. |
| \item |
\item |
| For~$n=1$, the condition $(\Delta f)(x)=f''(x)\equiv 0$ forces~$f$ to be an affine function on every segment; there are no ``interesting'' harmonic functions in one dimension. |
For~$n=1$, the condition $(\Delta f)(x)=f''(x)\equiv 0$ forces~$f$ to be an affine function on every segment; there are no ``interesting'' harmonic functions in one dimension. |
| \end{itemize} |
\end{itemize} |
