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.