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Oblique Derivative Problem

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

Abstract

Having proved in Chapter 9 that the Dirichlet problem on a spherical chip subject to a normal derivative condition on the flat portion of the boundary and a Dirichlet type condition on the remaining portion has a solution, it is shown that such a solution can be morphed onto a local solution of an elliptic equation on a neighborhood of a boundary point satisfying mixed boundary conditions. The Perron-Wiener-Brelot method is then used to prove the existence of a global solution on a bounded open subset of \(R^n\) that satisfies an oblique derivative boundary condition on a relatively open subset of the boundary and Dirichlet type condition on the remainder of the boundary. The usual criteria for regularity at points of the latter part of the boundary apply but the border points of the two types of subsets require a more restrictive “wedge condition” rather than a Zaremba cone condition. The chapter concludes with a theorem that establishes that the range of the resolvent of the elliptic operator is sufficient for applying the Hille-Yosida Theorem in the next chapter.

Keywords

Oblique Derivative Problem Dirichlet-type Condition Wedge Condition Regular Boundary Point Superfunction 
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.
    Hopf, E.: A remark on linear elliptic differential equations of second order. Proc. Amer. Math. Soc. 3, 791–793 (1952)Google Scholar
  2. 2.
    Lieberman, G.M.: The Perron process applied to oblique derivative problems. Adv. Math. 55, 161–172 (1985)Google Scholar
  3. 3.
    Apostol, T.M.: Mathematical Analysis. Addison-Wesley Publishing Company, Reading (1957)Google Scholar
  4. 4.
    Marsden, J.E., Tromba, A.J.: Vector Calculus, 2nd edn. W.H. Freeman and Co., New York (1981)Google Scholar
  5. 5.
    Lieberman, G.M.: Mixed boundary problems for elliptic and parabolic differential equations of second order. J. Math. Anal. Appl. 113, 422–440 (1986)Google Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.University of IllinoisUrbanaUSA

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