Laplace-Beltrami equation

Beltrami equation

A generalization of the Laplace equation for functions in a plane to the case of functions on an arbitrary two-dimensional Riemannian manifold of class . For a surface with local coordinates and first fundamental form

the Laplace–Beltrami equation has the form

(*)

For and , that is, when are isothermal coordinates on , equation (*) becomes the Laplace equation. The Laplace–Beltrami equation was introduced by E. Beltrami in 1864–1865 (see [1]).

The left-hand side of equation (*) divided by is called the second Beltrami differential parameter.

Regular solutions of the Laplace–Beltrami equation are generalizations of harmonic functions and are usually called harmonic functions on the surface (cf. also Harmonic function). These solutions are interpreted physically like the usual harmonic functions, e.g. as the velocity potential of the flow of an incompressible liquid flowing over the surface , or as the potential of an electrostatic field on , etc. Harmonic functions on a surface retain the properties of ordinary harmonic functions. A generalization of the Dirichlet principle is valid for them: Among all functions of class in a domain that take the same values on the boundary as a harmonic function , the latter gives the minimum of the Dirichlet integral

where

is the first Beltrami differential parameter, which is a generalization of the square of the gradient to the case of functions on a surface.

For generalizations of the Laplace–Beltrami equation to Riemannian manifolds of higher dimensions see Laplace operator.

References

[1]	E. Beltrami, "Richerche di analisi applicata alla geometria" , Opere Mat. , 1 , Milano (1902) pp. 107–198
[2]	M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)