View source for Laplace-Beltrami equation

''Beltrami equation''

A generalization of the [[Laplace equation|Laplace equation]] for functions in a plane to the case of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574501.png" /> on an arbitrary two-dimensional Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574502.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574503.png" />. For a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574504.png" /> with local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574505.png" /> and [[First fundamental form|first fundamental form]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574506.png" /></td> </tr></table>

the Laplace–Beltrami equation has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l0574509.png" />, that is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745010.png" /> are [[Isothermal coordinates|isothermal coordinates]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745011.png" />, equation (*) becomes the Laplace equation. The Laplace–Beltrami equation was introduced by E. Beltrami in 1864–1865 (see [[#References|[1]]]).

The left-hand side of equation (*) divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745012.png" /> is called the second Beltrami differential parameter.

Regular solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745013.png" /> of the Laplace–Beltrami equation are generalizations of harmonic functions and are usually called harmonic functions on the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745014.png" /> (cf. also [[Harmonic function|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745015.png" />, or as the potential of an electrostatic field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745016.png" />, etc. Harmonic functions on a surface retain the properties of ordinary harmonic functions. A generalization of the [[Dirichlet principle|Dirichlet principle]] is valid for them: Among all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745017.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745018.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745019.png" /> that take the same values on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745020.png" /> as a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745021.png" />, the latter gives the minimum of the Dirichlet integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745022.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745023.png" /></td> </tr></table>

is the first Beltrami differential parameter, which is a generalization of the square of the gradient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745024.png" /> to the case of functions on a surface.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Beltrami,   "Richerche di analisi applicata alla geometria" , ''Opere Mat.'' , '''1''' , Milano  (1902)  pp. 107–198</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Schiffer,   D.C. Spencer,   "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR></table>