A counterexample to the $L^{p}$-Hodge decomposition

• Autorzy: Piotr Hajłasz
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

nonuniqueness; Dirichlet problem; Laplace equation; Hodge decomposition

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1996
• Tom: 33
• Numer: 1
• Strony: 79-83

Daty

wydano

1996

Twórcy

autor

Piotr Hajłasz

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

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