On the divergence problem in some particular domains

For each function f ∈ L^p (Ω), 1 < p < ∞, with a vanishing mean value over a bounded Lipschitz domain Ω of Rⁿ, the equation div u = f has a solution in (W₀^1,p(Ω))ⁿ whose W^1,p-norm is bounded from above by the L^p-norm of f multiplied by a constant C independent of f. While this existence result is well-known, the estimates of the (best) constant C in terms of the domain Ω are rough. We study here the above problem in the particular case where the domain Ω is of the form A_l x Ω, where l is a parameter going to infinity, Ω is a bounded Lipschitz domain, and A_l is either an open ball with radius l, or a tubular annuli with constant thickness and interior radius l. We establish in particular that, in both cases, the corresponding constant C blows up as l goes to infinity.