Regularity for entropy solutions of parabolic \(p\)-Laplacian type equations (Q1576600)

In this elegant paper the authors obtain estimates in Marcinkiewicz spaces for the solution \(u\) to equations of the type of the parabolic \(p\)-Laplacian \(u_t - \mathop{div} a_p (x, \nabla u(x))=f,\) and its spatial gradient \(\nabla u\). The assumptions are \(f \in L^1 (\Omega \times (0,T),\) \(u_0 \in L^1(\Omega),\) where \(\Omega \subset \mathbb R^n\) may be bounded or unbounded, and \(a_p\) is a Carathéodory function satisfying the Leray-Lions conditions, a model example of \(a_p\) would be the function \(|\xi |^{p-2} \xi.\) Estimates for both cases \(p>2N/(N+1)\) and \(1<p<2N/(N+1)\) are proved. These results improve and notoriously extend those obtained in [\textit{L. Boccardo}, \textit{A. Dall'Aglio}, \textit{T. Gallouët} and \textit{L. Orsina}, J. Funct. Anal. 147, No. 1, 237--258 (1997; Zbl 0887.35082)]. This extension is achieved thanks to the Marcinkiewicz spaces setting: the results in the cited reference hold in \(L^p\) and Sobolev spaces, and depend crucially on the boundedness of the domain \(\Omega\) and restrictions on \(p.\) The authors point out that Marcinkiewicz spaces seem to be the natural context for the regularity problem for solutions to \(p\)-Laplacian type equation with data in \(L^1\).