On the divergence problem in some particular domains

摘要

For each function f is an element of L-p(Omega), 1< p < infinity, with a vanishing mean value over a bounded Lipschitz domain Omega of R-n, the equation div u=f has a solution in (W-0(1,p) (Omega))(n) whose W-1,W-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 Omega are rough. We study here the above problem in the particular case where the domain Omega is of the form A(l) x omega, where l is a parameter going to infinity, omega 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.