Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron Kerman; Luboš Pick

Ron Kerman ; Luboš Pick

Studia Mathematica, Tome 204 (2011), p. 97-119 / Harvested from The Polish Digital Mathematics Library

Résumé

We study imbeddings of the Sobolev space $W^{m, ϱ} (Ω)$ : = u: Ω → ℝ with $ϱ (\partial^{α} u / \partial x^{α})$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, $τ_{ϱ}$ and $σ_{ϱ}$ , that are optimal with respect to the inclusions $W^{m, ϱ} (Ω) \subset W^{m, τ_{ϱ}} (Ω) \subset L_{σ_{ϱ}} (Ω)$ . General formulas for $τ_{ϱ}$ and $σ_{ϱ}$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

Publié le : 2011-01-01

Zbl 1238.46026

EUDML-ID : urn:eudml:doc:285787

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     author = {Ron Kerman and Lubo\v s Pick},
     title = {Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities},
     journal = {Studia Mathematica},
     volume = {204},
     year = {2011},
     pages = {97-119},
     zbl = {1238.46026},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-2-1}
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Ron Kerman; Luboš Pick. Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities. Studia Mathematica, Tome 204 (2011) pp. 97-119. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm206-2-1/