A proof of the Faber-Krahn inequality for the first eigenvalue of thep-Laplacian
- 325 Downloads
In this work we present an elementary proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian on bounded domains in ℝn. Let λ1 be the first eigenvalue and λ 1 * be the first eigenvalue for the ball of the same volume. Then we show that λ1=λ 1 * iff the domain is a ball. Our proof makes considerable use of the corresponding Talenti's inequality and some well known properties of the first eigenfunction.
KeywordsElliptic Equation Maximal Function Measure Zero Orlicz Space Isoperimetric Inequality
Unable to display preview. Download preview PDF.
- C. Bandle,Isoperimetric Inequalities and Applications, Pitman Monographs and Studies in Math.,7, Boston (1980).Google Scholar
- T. Bhattacharya,A note on the Faber-Krahn inequality, Le Matematiche, vol.LIII, Fasc. 1 (1998), pp. 71–83.Google Scholar
- T. Bhattacharya—A. Weitsman,Some estimates for the symmetrized first eigenfunction of the Laplacian, to appear in J. Potential Anal.Google Scholar
- J. Brothers—W. Ziemer,Minimal rearrangements of Sobolev functions, J. Reine Angew. Math.,384 (1988).Google Scholar
- Yu. D. Burago—V. A. Zalgaller,Geometric Inequalities, Springer Verlag.Google Scholar
- D. Gilbarg—N. S. Trudinger,Elliptic Partial Differential equations of Second Order. Springer Verlag (1983).Google Scholar
- G. M. Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal. TMA, vol.12, no. 11, pp. 1203–1219.Google Scholar
- P. Lindqvist,On the equation div(|Du| p−2 Du)+λ|u| p−2 u=0, Proc. AMS. vol.109, no. 1 (1990).Google Scholar
- P. Lindqvist, Addendum to ≪On the equation div(|Du| p−2 Du)+λ|u| p−2 u=0≫, Proc. AMS., vol.112, no. 2 (1992).Google Scholar