Concave solutions of a Dirichlet problem
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- by H. Guggenheimer PDF
- Proc. Amer. Math. Soc. 40 (1973), 501-506 Request permission
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
- T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. MR 0344997
- P. R. Garabedian, Partial differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0162045
- 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 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)
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 40 (1973), 501-506
- MSC: Primary 31A20
- DOI: https://doi.org/10.1090/S0002-9939-1973-0330481-7
- MathSciNet review: 0330481