On the divergence problem in some particular domains

Abstract

For each function $f\in L^p(\Omega )$, $1<p<\infty$, with a vanishing mean value over a bounded Lipschitz domain $\Omega$ of ${{\mathbb {R}}}^n$, the equation $\mathrm{div}\, u = f$ has a solution in $(W^{1,p}_0(\Omega ))^n$ 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 $\Omega$ are rough. We study here the above problem in the particular case where the domain $\Omega$ is of the form $A_\ell \times \omega$, where $\ell$ is a parameter going to infinity, $\omega$ is a bounded Lipschitz domain, and $A_\ell$ is either an open ball with radius $\ell$, or a tubular annuli with constant thickness and interior radius $\ell$. We establish in particular that, in both cases, the corresponding constant C blows up as $\ell$ goes to infinity.