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<p>The aim of this article is to study a variant of the Poincaré inequality in Sobolev spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(W^{k,p(\cdot )}(\Omega ) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a given open set of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {R}^{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with finite Lebesgue measure and, consequently, the Friedrichs type inequality.</p>

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Note on the Poincaré and Friedrichs Inequalities in Sobolev Spaces with Variable Exponents

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  • 发表日期：

    2025年11月11日

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  • E. Viszus

摘要

The aim of this article is to study a variant of the Poincaré inequality in Sobolev spaces \(W^{k,p(\cdot )}(\Omega ) \) W k , p ( · ) ( Ω ) where \(\Omega \) Ω is a given open set of \(\mathbb {R}^{n}\) R n with finite Lebesgue measure and, consequently, the Friedrichs type inequality.