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arXiv · 1307.1340

Solvability of the divergence equation implies John via Poincaré inequality

Renjin Jiang·Aapo Kauranen·Pekka Koskela
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Abstract

Let $Ω\subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(Ω)^2$ for every $f\in L^p_0(Ω)$, if and only if $Ω$ is a John domain, if and only if the weighted Poincaré inequality $$\int_Ω|u(x)-u_Ω|^q\,dx\le C\int_Ω|\nabla u(x)|^q\dist(x,\partial Ω)^q\,dx$$ holds for some (every) $q\in [1,\fz)$. In higher dimensions similar results are proved under some additional assumptions on the domain in question.

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Renjin Jiang, Aapo Kauranen, Pekka Koskela. 2013-07-04. Solvability of the divergence equation implies John via Poincaré inequality. https://arxiv.org/abs/1307.1340

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