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A note on \(W^{1,p}\) estimates for quasilinear parabolic equations. (English) Zbl 1025.35007

The authors study interior \(W^{1,p}\) estimates for weak solutions to quasilinear parabolic equations in divergence form \[ u_t - \operatorname {div}a(x,t, \nabla u) = 0 \] where \(a\) satisfies some hypotheses, in particular that it is a Carathéodory function, i.e. that \(a(x,t,\zeta)\) in \((x,t)\) is measurable and it is continuous with respect to \(\zeta\) for each \(x.\)
They extend the elliptic case studied by L. Caffarelli and the first author in [Commun. Pure Appl. Math. 51, 1-21 (1998; Zbl 0906.35030)] to the parabolic case. The basic argument is to estimate the level sets of \(\nabla_x u\) and obtain the requested growth of their measure to have the integrability property. An important tool to obtain this results is a form of the Calderon-Zygmund covering property.

MSC:

35B45 A priori estimates in context of PDEs
35K55 Nonlinear parabolic equations
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0906.35030
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