Advertisement

Dirichlet Problem for Unbounded Regions

  • Lester L. HelmsEmail author
Chapter
  • 2.5k Downloads
Part of the Universitext book series (UTX)

Abstract

In this chapter, a study of the Dirichlet problem on unbounded regions is undertaken that necessitates enlarging \(R^n\) to \(R^n_{\infty }\) to include the point at infinity. Harmonic and superharmonic functions are defined anew on \(R^n_{\infty }\) to deal with the behavior of these functions at \(\infty \). A Poisson type integral is developed to solve the Dirichlet problem on the exterior of a ball and then to show that the Perron-Weiner-Brelot method can be used to prove the existence of a solution to the Dirichlet problem on unbounded regions. It is shown that the Poincaré and Zaremba criteria for regularity is applicable to finite boundary points and it is shown that \(\infty \) is always a regular boundary point for an unbounded region in \(R^n_{\infty }, n \ge 3\). A concept of thinness is used to characterize regular boundary points. A topology based on the concept of thinness is defined which is more natural from the potential theory view than the usual metric topology.

Keywords

Unbounded Region Superharmonic Function Regular Boundary Point Poisson Type Integrals Polar Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Brelot, M.: Sur le rôle du point à l’infini dans la théorie des fonctions harmoniques. Ann. Sci. École Norm. Sup. 61, 301–332 (1944)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Schwarz, H.A.: Gesammelte Mathematische Abhandlunger, vol. II. Springer, Berlin (1890)Google Scholar
  3. 3.
    Kellogg, O.D.: Foundations of Potential Theory. Dover Publications Inc, Berlin (1929) (reprinted by Dover, New York, 1953).Google Scholar
  4. 4.
    Bouligand, G.: Sur les fonctions bornée et harmoniques dans un domaine infini, nulles sur sa frontière. C.R. Acad. Sci. 169, 763–766 (1919)Google Scholar
  5. 5.
    Brelot, M.: Sur les ensembles effilés. Bull. Math. Soc. France 68, 12–36 (1944)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.University of IllinoisUrbanaUSA

Personalised recommendations