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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152601.png" /> be a domain in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152603.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152604.png" /> be a point on its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152605.png" />. A barrier for the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152606.png" /> is any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152607.png" />, continuous in the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152608.png" /> of the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b0152609.png" /> with some ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526010.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526011.png" />, which is superharmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526012.png" /> and positive in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526013.png" />, except at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526014.png" />, at which it vanishes. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526016.png" /> is any boundary point for which there exists a closed ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526018.png" /> which meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526019.png" /> only in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526020.png" />, one can take as a barrier the harmonic function

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526021.png" />

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526022.png" /> is the radius of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526024.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526025.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526026.png" /> is a domain of holomorphy. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526027.png" /> be a domain in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526029.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526030.png" /> be a point of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526031.png" />. Any analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526033.png" /> with a singular point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526034.png" /> will then be a barrier at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526035.png" />. Thus, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526036.png" /> is a barrier for the boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526037.png" /> of any plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526038.png" />. There also exists a barrier at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526039.png" /> of the boundary of the ball

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526040.png" />

e.g. the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526041.png" />.

A barrier exists at a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526042.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526043.png" /> if there is an analytic function defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526044.png" /> that is unbounded at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526045.png" />, i.e. is such that for some sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526046.png" /> which converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526047.png" /> one has:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526048.png" />

The converse is true for domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526049.png" /> in the following strong form: For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526050.png" /> of boundary points of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526051.png" /> at which a barrier exists, one can find a function holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526052.png" /> which is unbounded at all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526054.png" /> is everywhere dense in the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526055.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015260/b01526056.png" /> is a domain of holomorphy.

References

[1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654 Template:ZBL
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) pp. Chapt. 3 (Translated from Russian) MR Template:ZBL
[3]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) pp. Chapt. 3 (In Russian) MR Template:ZBL Template:ZBL

Comments

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

References

[a1]	W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Cambridge Univ. Press (1976) MR0460672 MR0419791 MR0412442 MR0442324 Template:ZBL Template:ZBL Template:ZBL
[a2]	L.L. Helms, "Introduction to potential theory" , Acad. Press (1975) (Translated from German) MR0460666 Template:ZBL