A counterexample to the L p -Hodge decomposition

Piotr Hajłasz

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 79-83
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Hajłasz, Piotr. "A counterexample to the $L^{p}$-Hodge decomposition." Banach Center Publications 33.1 (1996): 79-83. <http://eudml.org/doc/262838>.

@article{Hajłasz1996,
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.},
author = {Hajłasz, Piotr},
journal = {Banach Center Publications},
keywords = {nonuniqueness; Dirichlet problem; Laplace equation; Hodge decomposition; nonexistence; -Hodge decomposition},
language = {eng},
number = {1},
pages = {79-83},
title = {A counterexample to the $L^\{p\}$-Hodge decomposition},
url = {http://eudml.org/doc/262838},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Hajłasz, Piotr
TI - A counterexample to the $L^{p}$-Hodge decomposition
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 79
EP - 83
AB - 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.
LA - eng
KW - nonuniqueness; Dirichlet problem; Laplace equation; Hodge decomposition; nonexistence; -Hodge decomposition
UR - http://eudml.org/doc/262838
ER -

References

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