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[parent] examples of harmonic functions on $\mathbb{R}^n$ (Example)

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?

  • For $ n\ge 3$, define (on the punctured space  $ U=\mathbb{R}^n \setminus \{0\}$) the function  $ f(x)=\left\Vert x\right\Vert^{2-n}$. Then
    $\displaystyle \frac{\partial f}{\partial x_i} = (2-n) \frac{x_i}{\left\Vert x\right\Vert^n}, $
    and
    $\displaystyle \frac{\partial^2 f}{{\partial x_i}^2} = n(n-2)\frac{x_i^2}{\left\Vert x\right\Vert^{n+2}} - (n-2)\frac{1}{\left\Vert x\right\Vert^n} $
    Summing over $ i=1,...,n$ shows $ \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$.
  • 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.



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Cross-references: dimension, segment, forces, derivation, plane, summing, harmonic functions, function, real functions

This is version 6 of examples of harmonic functions on $\mathbb{R}^n$, born on 2002-06-05, modified 2004-08-09.
Object id is 3041, canonical name is ExamplesOfHarmonicFunctionOnRn.
Accessed 2927 times total.

Classification:
AMS MSC: 31A05 (Potential theory :: Two-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
 31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
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