Harmonic majorant

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'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)