<p>We consider vector valued weak solutions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\(u:\Omega _T\rightarrow \mathbb {R}^N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>:</mo> <msub> <mi mathvariant="normal">Ω</mi> <mi>T</mi> </msub> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(N\in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> of degenerate or singular parabolic systems of type <Equation ID="Equ164"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_Equ164.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="377" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \partial _t u - \textrm{div} \, a(z,u,Du) = 0 \qquad \text {in}\qquad \Omega _T= \Omega \times (0,T), \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>-</mo> <mtext>div</mtext> <mspace width="0.166667em" /> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mtext>in</mtext> <mspace width="2em" /> <msub> <mi mathvariant="normal">Ω</mi> <mi>T</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is an open set in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <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> for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(T&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> denotes a finite time. Assuming that the vector field <i>a</i> is not of Uhlenbeck-type structure, is continuous with respect to the solution variable <i>u</i> and satisfies a VMO-condition in the space-time variable <i>z</i>, we show that the solution <i>u</i> is partially Hölder continuous for every exponent <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1113_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, provided the vector field degenerates like that of the <i>p</i>-Laplacian for small gradients.</p>

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Hölder continuity for systems with degenerate diffusion and VMO-coefficients

  • Fabian Bäuerlein

摘要

We consider vector valued weak solutions \(u:\Omega _T\rightarrow \mathbb {R}^N\) u : Ω T R N with \(N\in \mathbb {N}\) N N of degenerate or singular parabolic systems of type \(\begin{aligned} \partial _t u - \textrm{div} \, a(z,u,Du) = 0 \qquad \text {in}\qquad \Omega _T= \Omega \times (0,T), \end{aligned}\) t u - div a ( z , u , D u ) = 0 in Ω T = Ω × ( 0 , T ) , where \(\Omega \) Ω is an open set in \(\mathbb {R}^{n}\) R n for \(n\ge 1\) n 1 and \(T>0\) T > 0 denotes a finite time. Assuming that the vector field a is not of Uhlenbeck-type structure, is continuous with respect to the solution variable u and satisfies a VMO-condition in the space-time variable z, we show that the solution u is partially Hölder continuous for every exponent \(\alpha \in (0,1)\) α ( 0 , 1 ) , provided the vector field degenerates like that of the p-Laplacian for small gradients.