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<p>We study the Calderón-Zygmund theory for nonlinear <i>p</i>-Laplacian type elliptic equations from composite material. We assume that the composite material is composed of several subdomains in the whole domain and the associated measurable nonlinearity under consideration is locally merely measurable in a variable but has small bounded mean oscillation in the other variables in each subdomain. From the relation between the internal geometry of composite material and measurable nonlinearities, we establish global <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3008_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(W^{1,q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> estimates for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3008_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">\(p\le q&lt;\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≤</mo> <mi>q</mi> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and as a corollary, we obtain the minimal regularity requirements of Calderón-Zygmund estimates for <i>p</i>-Laplacian type elliptic equations from the internal structure of composite material.</p>

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Gradient estimates for nonlinear equations with measurable nonlinearities from composite material

  • 期刊论文

  • 发表日期：

    2025年5月5日

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  • Yunsoo Jang

摘要

We study the Calderón-Zygmund theory for nonlinear p-Laplacian type elliptic equations from composite material. We assume that the composite material is composed of several subdomains in the whole domain and the associated measurable nonlinearity under consideration is locally merely measurable in a variable but has small bounded mean oscillation in the other variables in each subdomain. From the relation between the internal geometry of composite material and measurable nonlinearities, we establish global \(W^{1,q}\) W 1 , q estimates for \(p\le q<\infty \) p ≤ q < ∞ and as a corollary, we obtain the minimal regularity requirements of Calderón-Zygmund estimates for p-Laplacian type elliptic equations from the internal structure of composite material.