The limit as of solutions to the inhomogeneous Dirichlet problem of the -Laplacian

✍ Scribed by Mayte Perez-Llanos; Julio D. Rossi

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 341 KB
• Volume: 73
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2010.05.032

✦ Synopsis

In this work, we study the behaviour of the solutions to the following Dirichlet problem related to the p(x)-Laplacian operator,

on ∂Ω, as p(x) → ∞, for some suitable functions f . We consider a sequence of functions p n (x) that goes to infinity uniformly in Ω. Under adequate hypotheses on the sequence p n , basically, that the following two limits exist, lim n→∞ ∇ ln p n (x) = ξ (x), and lim sup n→∞ max x∈Ω p n min x∈Ω p n

≤ k, for some k > 0, we prove that u pn → u ∞ uniformly in Ω. In addition, we find that u ∞ solves a certain partial differential equation (PDE) problem (that depends on f ) in the viscosity sense. In particular, when f ≡ 1 in Ω, we get u ∞ (x) = dist(x, ∂Ω), and it turns out that the limit equation is |∇u| = 1.