View source for Harmonic majorant

''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" />''

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.

<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>

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:

<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>

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.

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" />.

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====
<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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR></table>