Laplace-Beltrami equation

Beltrami equation

A generalization of the Laplace equation for functions in a plane to the case of functions $ u $ on an arbitrary two-dimensional Riemannian manifold $ R $ of class $ C ^ {2} $. For a surface $ R $ with local coordinates $ \xi , \eta $ and first fundamental form

$$ d s ^ {2} = E d \xi ^ {2} + 2 F d \xi d \eta + G d \eta ^ {2} , $$

the Laplace–Beltrami equation has the form

$$ \tag{* } \Delta u \equiv \ \frac \partial {\partial \xi } \left ( \frac{F \frac{\partial u }{\partial \eta } - G \frac{\partial u }{\partial \xi } }{\sqrt {E G - F ^ { 2 } } } \right ) + \frac \partial {\partial \eta } \left ( \frac{F \frac{\partial u }{\partial \xi } - E \frac{\partial u }{\partial \eta } }{\sqrt {E G - F ^ { 2 } } } \right ) = 0 . $$

For $ E = G $ and $ F = 0 $, that is, when $ ( \xi , \eta ) $ are isothermal coordinates on $ R $, 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 $ \sqrt {E G - F ^ { 2 } } $ is called the second Beltrami differential parameter.

Regular solutions $ u $ of the Laplace–Beltrami equation are generalizations of harmonic functions and are usually called harmonic functions on the surface $ R $( 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 $ R $, or as the potential of an electrostatic field on $ R $, 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 $ v $ of class $ C ^ {2} ( G) \cap C ( \overline{G}\; ) $ in a domain $ G \subset R $ that take the same values on the boundary $ \partial G $ as a harmonic function $ v \in C ( \overline{G}\; ) $, the latter gives the minimum of the Dirichlet integral

$$ D ( v) = {\int\limits \int\limits } _ { G } \nabla v \cdot \sqrt {E G - F ^ { 2 } } \ d \xi d \eta , $$

where

$$ \nabla v = \ \frac{E \left ( \frac{\partial v }{\partial \eta } \right ) ^ {2} - 2 F \frac{\partial v }{\partial \xi } \frac{\partial v }{\partial \eta } + G \left ( \frac{\partial v }{\partial \xi } \right ) ^ {2} }{E G - F ^ { 2 } } $$

is the first Beltrami differential parameter, which is a generalization of the square of the gradient $ \mathop{\rm grad} ^ {2} u $ 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)