Harmonic majorant
least harmonic majorant of a family
The lower envelope of the family of all superharmonic majorants
of the family
of subharmonic functions on an open set
of a Euclidean space
,
, i.e.
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The least harmonic majorant is either a harmonic function or
on
. If the family consists of a single function
which is subharmonic on a larger set
, the concept of the best harmonic majorant
— the solution of the generalized Dirichlet problem for
with value
on the boundary
— may be employed. Always
, and the following formula [1] is valid:
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where is the measure which is associated with
,
, and
is the (generalized) Green function of the Dirichlet problem for
. The best and the least harmonic majorants coincide if and only if the set of all irregular points (cf. Irregular boundary point) of
has
-measure zero.
Correspondingly, if is a family of superharmonic functions on
, the greatest harmonic minorant
of the family
is defined as the upper envelope of the family of all subharmonic minorants of
; here
is the least harmonic majorant for
.
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'equilibre 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) |
Harmonic majorant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_majorant&oldid=47181