On the torsion function with Robin or Dirichlet boundary conditions

For p ∈ (1,+∞) and b ∈ (0,+∞] the p-torsion function with Robin boundary conditions associated to an arbitrary open set Ω ⊂ R satisfies formally the equation −∆p = 1 in Ω and |∇u|p−2 ∂u ∂n + b|u| p−2u = 0 on ∂Ω. We obtain bounds of the L∞ norm of u only in terms of the bottom of the spectrum (of the Robin p-Laplacian), b and the dimension of the space in the following two extremal cases: the linear framework (corresponding to p = 2) and arbitrary b > 0, and the non-linear framework (corresponding to arbitrary p > 1) and Dirichlet boundary conditions (b = +∞). In the general case, p 6= 2, p ∈ (1,+∞) and b > 0 our bounds involve also the Lebesgue measure of Ω. Mathematics Subject Classification (2000): 35J25, 35P99, 58J35.