Marcinkiewicz estimates for degenerate parabolic equations with measure data (Q457649)

The author proves integrability results for solutions of degenerate parabolic equations of the form NEWLINE\[NEWLINE u_t - \operatorname{div}a(x,t,Du) = \mu NEWLINE\]NEWLINE in \(\Omega_T = \Omega\times (-T,0)\), where a bounded open set \(\Omega \subset R^n\), \(n \geq 2\), and the vector field \(a(\cdot)\) satisfies only minimal measurability and monotonicity assumptions of Ladyzhenskaya-Uralt'seva type. Here, \(\mu\) is a signed Borel measure with finite total mass, which in general does not belong to the dual of the energy space naturally associated with the operator on the left-hand side of the above-mentioned equations. The most prominent model is the degenerate parabolic \(p\)-Laplace equation with measurable coefficients.NEWLINENEWLINEThe object of this investigation is the improvement of the integrability for the gradient of solutions of the equation in the case when the measure on the right-hand side satisfies a quantitative density condition that, roughly speaking, says that the measure does not concentrate on sets with small parabolic Hausdorff dimension. The improved integrability of the gradient is formulated in terms of Marcinkiewicz spaces \(M^m\) with the norm NEWLINENEWLINE\[NEWLINE \sup_{\lambda > 0}\lambda^m|\{z \in \Omega_T\,:\, |f(z) > \lambda\}| = \|f\|^m_{M^m(\Omega_T)} < \infty. NEWLINE\]NEWLINE NEWLINEAs the main result, the author proves that, if NEWLINENEWLINE\[NEWLINE \mu \in L^{1,\vartheta}(\Omega_T), \quad \vartheta_c < \vartheta \leq N, NEWLINE\]NEWLINE NEWLINEthen NEWLINENEWLINE\[NEWLINE |Du| \in M^{p-1 + 1/(\vartheta - 1)}_{\text{loc}}(\Omega_T). NEWLINE\]