Weighted inequalities for gradients on non-smooth domains

Caroline Sweezy, and J. Michael Wilson. Weighted inequalities for gradients on non-smooth domains. 2010. <http://eudml.org/doc/285995>.

@book{CarolineSweezy2010,
abstract = {We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, $u|_\{∂Ω\} = f$, with f belonging to a reasonable test class, then $(∫_\{Ω\} |∇u|^\{q\} dμ) ^\{1/q\} ≤ (∫_\{∂Ω\} |f|^\{p\} dν)^\{1/p\}$, where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ₊^\{d+1\}$. As in that case we attack the problem by means of Littlewood-Paley theory. However, the lack of translation invariance forces us to use a general result of Wilson, which must then be translated into the setting of homogeneous spaces. We also consider what can be proved when a strictly elliptic divergence form operator replaces the Laplacian.},
author = {Caroline Sweezy, J. Michael Wilson},
keywords = {elliptic equations; boundary value problems; weighted norm inequalities; Littlewood-Paley theory; Lipschitz domains},
language = {eng},
title = {Weighted inequalities for gradients on non-smooth domains},
url = {http://eudml.org/doc/285995},
year = {2010},
}

TY - BOOK
AU - Caroline Sweezy
AU - J. Michael Wilson
TI - Weighted inequalities for gradients on non-smooth domains
PY - 2010
AB - We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, $u|_{∂Ω} = f$, with f belonging to a reasonable test class, then $(∫_{Ω} |∇u|^{q} dμ) ^{1/q} ≤ (∫_{∂Ω} |f|^{p} dν)^{1/p}$, where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ₊^{d+1}$. As in that case we attack the problem by means of Littlewood-Paley theory. However, the lack of translation invariance forces us to use a general result of Wilson, which must then be translated into the setting of homogeneous spaces. We also consider what can be proved when a strictly elliptic divergence form operator replaces the Laplacian.
LA - eng
KW - elliptic equations; boundary value problems; weighted norm inequalities; Littlewood-Paley theory; Lipschitz domains
UR - http://eudml.org/doc/285995
ER -