Harmonic majorant

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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)