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Newtonian Potential

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

Abstract

This chapter commences with the study of subnewtonian kernels that share many of the properties of the Newtonian kernel \(1/|x-y|^{n-2}\) on \(R^n \times R^n\) related to transforming a given function into another and the determination of properties of the transformed function passed on through the kernel. Poisson’s equation \(\triangle u = f\) is shown to be solvable if \(f\) has adequate smoothness properties. The last section introduces Schwarz’s Reflection Principle that allows the extension of function on one side of region \(\Omega \) that is symmetric relative to a hyperplane to all of \(\Omega \) with a vanishing normal derivative at points of the hyperplane that are in \(\Omega \).

Keywords

Newtonian Potential Reflection Principle Fundamental Harmonic Function Regular Boundary Point Nonhomogeneous Conditions 
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

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    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  2. 2.
    Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  3. 3.
    Marsden, J.E., Tromba, A.J.: Vector Calculus, 2nd edn. W.H. Freeman and Co., New York (1981)zbMATHGoogle Scholar
  4. 4.
    Hopf, E.: A remark on linear elliptic differential equations of second order. Proc. Amer. Math. Soc. 3, 791–793 (1952)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.University of IllinoisUrbanaUSA

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