Barrier

Lebesgue barrier, in potential theory

A function the existence of which is a necessary and sufficient condition for the regularity of a boundary point with respect to the behaviour of a generalized solution of the Dirichlet problem at that point (cf. Perron method; Regular boundary point).

Let  be a domain in a Euclidean space , , and let  be a point on its boundary . A barrier for the point  is any function , continuous in the intersection  of the closed domain  with some ball  with centre at , which is superharmonic in  and positive in , except at , at which it vanishes. For instance, if  and  is any boundary point for which there exists a closed ball  in  which meets  only in , one can take as a barrier the harmonic function

where  is the radius of  and  is its centre.

A barrier in the theory of functions of (several) complex variables is a function the existence of which for all boundary points of the domain  implies that  is a domain of holomorphy. Let  be a domain in the complex space , , and let  be a point of the boundary . Any analytic function  in  with a singular point at  will then be a barrier at . Thus, the function  is a barrier for the boundary point  of any plane domain . There also exists a barrier at any point  of the boundary of the ball

e.g. the function .

A barrier exists at a boundary point  of a domain  if there is an analytic function defined in  that is unbounded at , i.e. is such that for some sequence of points  which converges to  one has:

The converse is true for domains  in the following strong form: For any set  of boundary points of a domain  at which a barrier exists, one can find a function holomorphic in  which is unbounded at all points of . If  is everywhere dense in the boundary of , then  is a domain of holomorphy.

#### References[edit]

[1] |  R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654

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[2] |  V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) pp. Chapt. 3 (Translated from Russian)
[3] |  B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) pp. Chapt. 3 (In Russian) Zbl 0578.32001 Zbl 0574.30001

#### Comments[edit]

Good English references for the Lebesgue barrier are [a1] and [a2].

#### References[edit]

[a1] |  W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Cambridge Univ. Press (1976) MR0460672 MR0419791 MR0412442 MR0442324 Zbl 0419.31001 Zbl 0339.31003 Zbl 0328.33011

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[a2] |  L.L. Helms, "Introduction to potential theory" , Acad. Press (1975) (Translated from German) MR0460666 Zbl 0188.17203