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 be a domain in a -dimensional Euclidean space that is bounded by a Jordan surface (in particular, a smooth or piecewise-smooth surface without self-intersections) containing a -dimensional subdomain of a -dimensional hyperplane . If the function is harmonic in , continuous on and equal to zero everywhere on , then can be extended as a harmonic function across into the domain that is symmetric to relative to , by means of the equality

where the points and are symmetric relative to .

2) Let be a domain of a -dimensional Euclidean space that is bounded by a Jordan surface containing a -dimensional subdomain of a -dimensional sphere of radius with centre at a point . If is harmonic in , continuous on and equal to zero everywhere on , then can be extended as a harmonic function across into the domain that is symmetric to relative to (i.e. obtained from by means of the transformation of inverse radii — inversions — relative to ). This continuation is achieved by means of the Kelvin transformation, taken with the opposite sign, of relative to , namely:

where , . Under the transformation of inverse radii relative to , the point is mapped to the point , in correspondence with

such that if , then belongs to the domain (where is given), and if , then .

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 .

Cf. also Schwarz symmetry theorem.