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Proceedings of the American Mathematical Society

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Concave solutions of a Dirichlet problem

Author: H. Guggenheimer
Journal: Proc. Amer. Math. Soc. 40 (1973), 501-506
MSC: Primary 31A20
MathSciNet review: 0330481
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Abstract: We define a function that may serve as the measure of the deviation of a plane convex domain from circular shape. Then we show that the Dirichlet problem of $ \Delta u + \lambda p(x,y)u = 0$ and vanishing boundary values on a plane, convex domain can have a concave solution only if an integral condition involving the deviation from circular shape is satisfied. A weak form of the condition is generalized to $ n$ dimensions.

References [Enhancements On Off] (What's this?)

  • [1] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. MR 0344997
  • [2] P. R. Garabedian, Partial differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0162045
  • [3] Branko Grünbaum, Measures of symmetry for convex sets, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 233–270. MR 0156259
  • [4] N. E. Joukovsky, Conditions of boundedness of integrals of equations $ ({d^2}y/d{x^2}) + py = 0$, Mat. Sb. 16 (1891/93), 582-591. (Russian)

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Article copyright: © Copyright 1973 American Mathematical Society