Published Paper

Inserted: 25 jun 2003
Last Updated: 19 dec 2005

Journal: ZAMP Z. Angew. Math. Phys.
Volume: 54
Year: 2003

Abstract:

We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the $p$--Laplace operator $\Delta_p u=\sum_i {\frac{\partial}{\partial x_i}} (\vert\nabla u \vert^{p-2}{\frac{\partial u}{\partial x_i}})$, the pseudo--$p$--Laplace operator $\tilde\Delta_p u=\sum_i {\frac{\partial}{\partial x_i}} (\vert {\frac{\partial u}{\partial x_i}} \vert^{p-2} {\frac{\partial u}{\partial x_i}} )$ and others. We derive the positivity of the first eigenfunction, simplicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities. In another chapter we address the question of symmetry for positive solutions to more general equations. Using a Pohozaev-type inequality and isoperimetric inequalities as well as convex rearrangement methods we generalize a symmetry result of Kesavan and Pacella. Our optimal domains are level sets of a convex function $H^o$. They have the so-called Wulff shape associated with $H$ and only in special cases they are Euclidean balls.

Keywords: eigenvalue, $p$-Laplace operator, pseudo-$p$-Laplace operator, isoperimetric inequality, Wulff theorem, Wulff shape, convex rearrangement,, Faber Krahn inequality, Payne Rayner inequality, Cheeger inequality

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