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''least harmonic majorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464801.png" /> of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464802.png" />''
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The lower envelope of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464803.png" /> of all superharmonic majorants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464804.png" /> of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464805.png" /> of subharmonic functions on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464806.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h0464808.png" />, i.e.
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{{TEX|done}}
  
<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/h/h046/h046480/h0464809.png" /></td> </tr></table>
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''least harmonic majorant  $  v $
 +
of a family  $  \{ u _ {i} \} $''
  
The least harmonic majorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648010.png" /> is either a harmonic function or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648012.png" />. If the family consists of a single function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648013.png" /> which is subharmonic on a larger set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648014.png" />, the concept of the best harmonic majorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648015.png" /> — the solution of the generalized Dirichlet problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648016.png" /> with value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648017.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648018.png" /> — may be employed. Always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648019.png" />, and the following formula [[#References|[1]]] is valid:
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The lower envelope of the family  $  \mathfrak B = \{ v _ {k} \} $
 +
of all superharmonic majorants  $  v _ {k} $
 +
of the family $  \{ u _ {i} \} $
 +
of subharmonic functions on an open set  $  D $
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of a Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,  
 +
i.e.
  
<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/h/h046/h046480/h04648020.png" /></td> </tr></table>
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$$
 +
v ( x)  = \inf  \{ {v _ {k} ( x) } : {
 +
v _ {k} \in \mathfrak B } \}
 +
,\ \
 +
x \in D.
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648021.png" /> is the measure which is associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648024.png" /> is the (generalized) Green function of the Dirichlet problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648025.png" />. The best and the least harmonic majorants coincide if and only if the set of all irregular points (cf. [[Irregular boundary point|Irregular boundary point]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648026.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648027.png" />-measure zero.
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The least harmonic majorant  $  v $
 +
is either a harmonic function or  $  v ( x) \equiv + \infty $
 +
on  $  D $.  
 +
If the family consists of a single function  $  u $
 +
which is subharmonic on a larger set  $  D _ {0} \supset \overline{D}\; $,  
 +
the concept of the best harmonic majorant  $  v  ^ {*} $—
 +
the solution of the generalized Dirichlet problem for $  D $
 +
with value  $  u $
 +
on the boundary  $  \Gamma = \partial  D $—
 +
may be employed. Always  $  v  ^ {*} - v \geq  0 $,
 +
and the following formula [[#References|[1]]] is valid:
  
Correspondingly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648028.png" /> is a family of superharmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648029.png" />, the greatest harmonic minorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648030.png" /> of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648031.png" /> is defined as the upper envelope of the family of all subharmonic minorants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648032.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648033.png" /> is the least harmonic majorant for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046480/h04648034.png" />.
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$$
 +
v  ^ {*} ( x) - v ( x)  = \
 +
- \int\limits _  \Gamma  G ( x, y)  d \mu ( y),\ \
 +
x \in D,
 +
$$
 +
 
 +
where  $  \mu $
 +
is the measure which is associated with  $  u $,  
 +
$  \mu \leq  0 $,
 +
and  $  G( x, y) $
 +
is the (generalized) Green function of the Dirichlet problem for  $  D $.  
 +
The best and the least harmonic majorants coincide if and only if the set of all irregular points (cf. [[Irregular boundary point|Irregular boundary point]]) of  $  \Gamma $
 +
has  $  \mu $-
 +
measure zero.
 +
 
 +
Correspondingly, if  $  \{ \widetilde{u}  _ {i} \} $
 +
is a family of superharmonic functions on $  D $,  
 +
the greatest harmonic minorant $  w $
 +
of the family $  \{ \widetilde{u}  _ {i} \} $
 +
is defined as the upper envelope of the family of all subharmonic minorants of $  \{ \widetilde{u}  _ {i} \} $;  
 +
here $  - w $
 +
is the least harmonic majorant for $  \{ - \widetilde{u}  _ {i} \} $.
  
 
The problem of harmonic majorants may also be posed in terms of the Cauchy problem for the Laplace equation. See [[Harmonic function|Harmonic function]].
 
The problem of harmonic majorants may also be posed in terms of the Cauchy problem for the Laplace equation. See [[Harmonic function|Harmonic function]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Frostman,   "Potentiel d'equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions"  ''Mett. Lunds Univ. Mat. Sem.'' , '''3'''  (1935)  pp. 1–118</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  O. Frostman, "Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions"  ''Mett. Lunds Univ. Mat. Sem.'' , '''3'''  (1935)  pp. 1–118</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====

Latest revision as of 06:35, 17 July 2024


least harmonic majorant $ v $ of a family $ \{ u _ {i} \} $

The lower envelope of the family $ \mathfrak B = \{ v _ {k} \} $ of all superharmonic majorants $ v _ {k} $ of the family $ \{ u _ {i} \} $ of subharmonic functions on an open set $ D $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, i.e.

$$ v ( x) = \inf \{ {v _ {k} ( x) } : { v _ {k} \in \mathfrak B } \} ,\ \ x \in D. $$

The least harmonic majorant $ v $ is either a harmonic function or $ v ( x) \equiv + \infty $ on $ D $. If the family consists of a single function $ u $ which is subharmonic on a larger set $ D _ {0} \supset \overline{D}\; $, the concept of the best harmonic majorant $ v ^ {*} $— the solution of the generalized Dirichlet problem for $ D $ with value $ u $ on the boundary $ \Gamma = \partial D $— may be employed. Always $ v ^ {*} - v \geq 0 $, and the following formula [1] is valid:

$$ v ^ {*} ( x) - v ( x) = \ - \int\limits _ \Gamma G ( x, y) d \mu ( y),\ \ x \in D, $$

where $ \mu $ is the measure which is associated with $ u $, $ \mu \leq 0 $, and $ G( x, y) $ is the (generalized) Green function of the Dirichlet problem for $ D $. The best and the least harmonic majorants coincide if and only if the set of all irregular points (cf. Irregular boundary point) of $ \Gamma $ has $ \mu $- measure zero.

Correspondingly, if $ \{ \widetilde{u} _ {i} \} $ is a family of superharmonic functions on $ D $, the greatest harmonic minorant $ w $ of the family $ \{ \widetilde{u} _ {i} \} $ is defined as the upper envelope of the family of all subharmonic minorants of $ \{ \widetilde{u} _ {i} \} $; here $ - w $ is the least harmonic majorant for $ \{ - \widetilde{u} _ {i} \} $.

The problem of harmonic majorants may also be posed in terms of the Cauchy problem for the Laplace equation. See Harmonic function.

References

[1] O. Frostman, "Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions" Mett. Lunds Univ. Mat. Sem. , 3 (1935) pp. 1–118
[2] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)

Comments

In axiomatic potential theory (cf. Potential theory, abstract) the equality of the best and the least harmonic majorant is connected to the domination principle (cf. Domination), see [a1], Chapt. 9.

References

[a1] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
How to Cite This Entry:
Harmonic majorant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_majorant&oldid=15227
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article