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''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]]
+
A generalization of the [[Laplace equation|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|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>
+
$$
 +
d s  ^ {2}  = E  d \xi  ^ {2} + 2 F  d \xi  d \eta + G  d \eta  ^ {2} ,
 +
$$
  
 
the Laplace–Beltrami equation has the form
 
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>
+
$$ \tag{* }
 +
\Delta u  \equiv \
 +
 
 +
\frac \partial {\partial  \xi }
 +
 
 +
\left (
  
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]]]).
+
\frac{F
 +
\frac{\partial  u }{\partial  \eta }
 +
- G
 +
\frac{\partial  u }{\partial  \xi }
 +
}{\sqrt {E G - F ^ { 2 } } }
  
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.
+
\right )
 +
+
 +
\frac \partial {\partial  \eta }
  
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
+
\left (
  
<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>
+
\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|isothermal coordinates]] on  $  R $,
 +
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  $  \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|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|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
 
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>
+
$$
 +
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057450/l05745024.png" /> to the case of functions on a surface.
+
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|Laplace operator]].
 
For generalizations of the Laplace–Beltrami equation to Riemannian manifolds of higher dimensions see [[Laplace operator|Laplace operator]].

Latest revision as of 22:15, 5 June 2020


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-Beltrami_equation&oldid=13891
This article was adapted from an original article by E.D. SolomentsevE.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article