A counterexample to the $L^{p}$-Hodge decomposition
Volume 33 / 1996
Banach Center Publications 33 (1996), 79-83
DOI: 10.4064/-33-1-79-83
Abstract
We construct a bounded domain $Ω ⊂ ℝ^2$ with the cone property and a harmonic function on Ω which belongs to $W_0^{1,p}(Ω)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no $L^p$-Hodge decomposition in $L^{p}(Ω,ℝ^2)$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in $W^{1,p}(Ω)$ for all p > 4.
