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<p>In this study, we approach the analysis of a degenerate nonlinear functional in one dimension, accommodating a degenerate weight <i>w</i>. Our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting <i>p</i>-growth for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1058_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">\(1&lt; p&lt;+\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>p</mi> <mo>&lt;</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. We adopt the approach developed in [<CitationRef CitationID="CR6">6</CitationRef>], where some assumptions like doubling or Muckenhoupt conditions are dropped. Our main tools consist of proving the validity of a weighted Poincaré inequality involving an auxiliary weight.</p>

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Relaxation for degenerate nonlinear functionals in the one-dimensional case

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月12日

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  • Valeria Chiadò Piat,
  • Virginia De Cicco,
  • Anderson Melchor Hernandez

摘要

In this study, we approach the analysis of a degenerate nonlinear functional in one dimension, accommodating a degenerate weight w. Our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting p-growth for \(1< p<+\infty \) 1 < p < + ∞ . We adopt the approach developed in [6], where some assumptions like doubling or Muckenhoupt conditions are dropped. Our main tools consist of proving the validity of a weighted Poincaré inequality involving an auxiliary weight.