Reflection principle

A generalization of the symmetry principle for harmonic functions to harmonic functions in an arbitrary number of independent variables. Some formulations of the reflection principle are as follows:

1) Let $G$ be a domain in a $k$-dimensional Euclidean space $( k \geq 1)$ that is bounded by a Jordan surface $\Gamma$ (in particular, a smooth or piecewise-smooth surface $\Gamma$ without self-intersections) containing a $( k- 1)$-dimensional subdomain $\sigma$ of a $( k- 1)$-dimensional hyperplane $L$. If the function $U( x _ {1} \dots x _ {k} )$ is harmonic in $G$, continuous on $G \cup \sigma$ and equal to zero everywhere on $\sigma$, then $U( x _ {1} \dots x _ {k} )$ can be extended as a harmonic function across $\sigma$ into the domain $G ^ {*}$ that is symmetric to $G$ relative to $L$, by means of the equality

$$U( x _ {1} ^ {*} \dots x _ {k} ^ {*} ) = - U( x _ {1} \dots x _ {k} ),$$

where the points $( x _ {1} ^ {*} \dots x _ {k} ^ {*} ) \in G ^ {*}$ and $( x _ {1} \dots x _ {k} ) \in G$ are symmetric relative to $L$.

2) Let $G$ be a domain of a $k$-dimensional Euclidean space $( k \geq 1)$ that is bounded by a Jordan surface $\Gamma$ containing a $( k- 1)$-dimensional subdomain $\sigma$ of a $( k- 1)$-dimensional sphere $\Sigma$ of radius $R > 0$ with centre at a point $M ^ {0} = ( x _ {1} ^ {0} \dots x _ {k} ^ {0} )$. If $U( x _ {1} \dots x _ {k} )$ is harmonic in $G$, continuous on $G \cup \sigma$ and equal to zero everywhere on $\sigma$, then $U( x _ {1} \dots x _ {k} )$ can be extended as a harmonic function across $\sigma$ into the domain $G ^ {*}$ that is symmetric to $G$ relative to $\Sigma$ (i.e. obtained from $G$ by means of the transformation of inverse radii — inversions — relative to $\Sigma$). This continuation is achieved by means of the Kelvin transformation, taken with the opposite sign, of $U$ relative to $\Sigma$, namely:

$$U( x _ {1} ^ {*} \dots x _ {k} ^ {*} ) =$$

$$= \ - \frac{R ^ {k- 2} }{r ^ {k- 2} } U \left ( x _ {1} ^ {0} + R ^ {2} \frac{x _ {1} ^ {*} - x _ {1} ^ {0} }{r ^ {2} } \dots x _ {k} ^ {0} + R ^ {2} \frac{x _ {k} ^ {*} - x _ {k} ^ {0} }{r ^ {2} } \right ) ,$$

where $( x _ {1} ^ {*} \dots x _ {k} ^ {*} ) \in G ^ {*}$, $r = \sqrt {( x _ {1} ^ {*} - x _ {1} ^ {0} ) ^ {2} + \dots + ( x _ {k} ^ {*} - x _ {k} ^ {0} ) ^ {2} }$. Under the transformation of inverse radii relative to $\Sigma$, the point $M ^ {*} = ( x _ {1} ^ {*} \dots x _ {k} ^ {*} )$ is mapped to the point $M( x _ {1} \dots x _ {k} )$, in correspondence with

$$x _ {1} - x _ {1} ^ {0} = R ^ {2} \frac{x _ {1} ^ {*} - x _ {1} ^ {0} }{r ^ {2} } , \dots, x _ {k} - x _ {k} ^ {0} = \ R ^ {2} \frac{x _ {k} ^ {*} - x _ {k} ^ {0} }{r ^ {2} } ,$$

such that if $M ^ {*} \in G ^ {*}$, then $M$ belongs to the domain $G$ (where $U$ is given), and if $M ^ {*} \in \sigma$, then $M = M ^ {*}$.

References

[1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)

Comments

In the non-Soviet literature, "reflection principle" refers also to the Riemann–Schwarz principle and its generalizations to $\mathbf C ^ {n}$.

Cf. also Schwarz symmetry theorem.