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Some properties of the first eigenvalue of the p(x)-laplacian on riemannian manifolds

The main result of the present paper establishes a stability property of the first eigenvalue of the associated problem which deals with the p(x)-Laplacian on Riemannian manifolds with Dirichlet boundary condition. | Turk J Math 33 (2009) , 351 – 358. ¨ ITAK ˙ c TUB doi:10.3906/mat-0806-13 Some properties of the first eigenvalue of the p(x)-laplacian on riemannian manifolds R. A. Mashiyev, G. Alisoy and S. Ogras Abstract The main result of the present paper establishes a stability property of the first eigenvalue of the associated problem which deals with the p(x) -Laplacian on Riemannian manifolds with Dirichlet boundary condition. Key Words: Variable exponent Lebesgue and Sobolev spaces; first eigenvalue; Riemannian manifolds; p(x) -Laplacian. 1. Introduction Over the last decades the variable exponent Lebesgue spaces Lp(x) and the corresponding Sobolev space W 1,p(x) have been a subject of active research stimulated by development of the studies of problems in elasticity, fluid dynamics, calculus of variations, and differential equations with p(x)-growth (see [2], [3], [12]). We refer the reader to [5], [7], [8] for fundamental properties of these spaces. The p(x)-Laplacian equations related to eigenvalue problems have been studied in [6], [9], [10], [11]. Let G ⊂ RN (N ≥ 2 ) is a bounded domain with a smooth boundary. For measurable function p(x) we denote the variable exponent Lebesgue space by Lp(x) (G) = ⎧ ⎨ ⎩ |u (x)| u measurable real functions : p(x) dx 0 : dx ≤ 1 , ⎩ ⎭ δ G where 1 1, then the spaces Lp(x) (G) , W 1,p(x) (G) and W0 spaces (see [5], [7]). Proposition 1. ([5], [7] ). Denote p(x) (u) |u (x)| = p(x) dx, G and 1,p(x) (u) := |∇u (x)| p(x) (∇u) = p(x) dx, ∀u, ∇u ∈ Lp(x) (G) , G then we have p− p+ min |u|p(x) , |u|p(x) ≤ p− p+ min |∇u|p(x) , |∇u|p(x) ≤ p(x) p− p+ (u) ≤ max |u|p(x) , |u|p(x) , 1,p(x) p− p+ (u) ≤ max |∇u|p(x) , |∇u|p(x) . M be a compact Riemannian manifold with dim M = m, and p(x) is nonhomegenous p(x) p(x)−2 Laplacian acting on functions on M, where p(x) u = div |∇u| ∇u , and 1 0 ; that is, Bε = {x ∈ M : d (x, M ∗) 0. p(x)−2 When Bε = ∅, that is, Ωε = M¨, u runs over W 1,p(x)