# Laplace-Beltrami equation

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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)
How to Cite This Entry:
Laplace–Beltrami equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace%E2%80%93Beltrami_equation&oldid=22708