<p>We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form <Equation ID="Equ55"> <EquationSource Format="TEX">\(\begin{aligned} \partial _t u^i - \textrm{div} \big ( a(|Du|) Du^i \big )= f^i, \qquad i=1,\dots ,N, \end{aligned}\)</EquationSource> </Equation>in a space-time cylinder <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Omega _T = \Omega \times (0,T)\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega \subset \mathbb {R}^n\)</EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n \ge 2\)</EquationSource> </InlineEquation>) is a bounded, convex <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C^2\)</EquationSource> </InlineEquation>-domain and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(T&gt;0\)</EquationSource> </InlineEquation>. The inhomogeneity <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(f=(f^1,\dots ,f^N)\)</EquationSource> </InlineEquation> belongs to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(L^{n+2+\sigma }(\Omega _T,\mathbb {R}^N)\)</EquationSource> </InlineEquation> for some <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\sigma &gt;0\)</EquationSource> </InlineEquation>. The coefficients <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(a:\mathbb {R}_{&gt;0} \rightarrow \mathbb {R}_{&gt;0}\)</EquationSource> </InlineEquation> are of Uhlenbeck-type and satisfy a nonstandard (<i>p</i>,&#xa0;<i>q</i>)-growth condition with <Equation ID="Equ56"> <EquationSource Format="TEX">\( 2 \le p \le q &lt; p + \frac{4}{n+2}. \)</EquationSource> </Equation>Our main result establishes a local Lipschitz estimate up to the lateral boundary for any local weak solution that vanishes on the lateral boundary of the cylinder.</p>

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Boundary regularity for parabolic systems with nonstandard (p,q)-growth conditions in smooth convex domains

  • Michael Strunk

摘要

We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form \(\begin{aligned} \partial _t u^i - \textrm{div} \big ( a(|Du|) Du^i \big )= f^i, \qquad i=1,\dots ,N, \end{aligned}\) in a space-time cylinder \(\Omega _T = \Omega \times (0,T)\) , where \(\Omega \subset \mathbb {R}^n\) ( \(n \ge 2\) ) is a bounded, convex \(C^2\) -domain and \(T>0\) . The inhomogeneity \(f=(f^1,\dots ,f^N)\) belongs to \(L^{n+2+\sigma }(\Omega _T,\mathbb {R}^N)\) for some \(\sigma >0\) . The coefficients \(a:\mathbb {R}_{>0} \rightarrow \mathbb {R}_{>0}\) are of Uhlenbeck-type and satisfy a nonstandard (pq)-growth condition with \( 2 \le p \le q < p + \frac{4}{n+2}. \) Our main result establishes a local Lipschitz estimate up to the lateral boundary for any local weak solution that vanishes on the lateral boundary of the cylinder.