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

Banach Center Publications (1996)

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

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topHajłasz, 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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