On the torsion function with Robin or Dirichlet boundary conditions

For p∈(1,+∞)p∈(1,+∞) and b∈(0,+∞]b∈(0,+∞] the p  -torsion function with Robin boundary conditions associated to an arbitrary open set Ω⊂RmΩ⊂Rm satisfies formally the equation −Δp=1−Δp=1 in Ω   and |∇u|p−2∂u∂n+b|u|p−2u=0 on ∂Ω  . We obtain bounds of the L∞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=2p=2) and arbitrary b>0b>0, and the non-linear framework (corresponding to arbitrary p>1p>1) and Dirichlet boundary conditions (b=+∞b=+∞). In the general case, p≠2p≠2, p∈(1,+∞)p∈(1,+∞) and b>0b>0 our bounds involve also the Lebesgue measure of Ω.