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New results of Hölder continuity for non variational basic parabolic systems. (English) Zbl 0933.35038

Summary: Let \(\Omega\) be a bounded open set of \(\mathbb{R}^n\) \((n\geq 2)\) and \(Q\) the cylinder \(\Omega\times(-T,0)\). We prove, for \(n\leq 6\), the Hölder-continuity in \(Q\) of the strong solutions \(u\) of the following nonlinear non-variational “basic” parabolic system \[ a\bigl(H(u)\bigr)-{\partial u\over \partial t}=0, \] where \(a(\xi)\) is a vector of \(\mathbb{R}^N\) \((N\) integer \(\geq 1)\), continuous onto \(\mathbb{R}^{n^2N}\), satisfying the condition \(a(0)=0\) and if there exist three positive constants \(\alpha,\gamma\), and \(\delta\), with \(\gamma+\delta<1\), such that \[ \left\|\sum^n_{i=1}\tau_{ii}-\alpha\bigl[a(\tau+\eta)-a(\eta)\bigr]\right\|^2\leq\gamma\|\tau\|^2+\delta\left\|\sum^n_{i=1}\tau_{ii} \right\|^2,\quad\forall\tau,\eta\in\mathbb{R}^{n^2N}. \] We show also the Hölder-continuity in \(Q\) of the gradient \(Du\) for \(n\leq 4\), of the matrix \(H(u)\) and of the vector \({\partial u\over\partial t}\) for \(n=2\).

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35K40 Second-order parabolic systems
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
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