Revista Matemática Iberoamericana

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Volume 30, Issue 4, 2014, pp. 1355–1386
DOI: 10.4171/RMI/817

Published online: 2014-12-15

Calderón–Zygmund estimates for parabolic $p(x, t)$-Laplacian systems

Paolo Baroni[1] and Verena Bögelein[2]

(1) Uppsala University, Sweden
(2) Universität Salzburg, Austria

We prove local Calderón–Zygmund estimates for weak solutions of the evolutionary $p(x,t)$-Laplacian system $$ \partial_t u-\mathrm {div}\ \big(a(x,t){|Du|}^{p(x,t)-2}Du\big) = \mathrm {div}\ \big({|F|}^{p(x,t)-2}F\big)$$ under the classical hypothesis of logarithmic continuity for the variable exponent $p(x,t)$. More precisely, we show that the spatial gradient $Du$ of the solution is as integrable as the right-hand side $F$, i.e., $$|F|^{p(\cdot)}\in L^q_\mathrm {loc} \ \Longrightarrow\ |Du|^{p(\cdot)}\in L^q_\mathrm {loc} \quad\mbox{for any $q>1$},$$ together with quantitative estimates. Thereby we allow the presence of eventually discontinuous coefficients $a(x,t)$, requiring only a VMO condition with respect to the spatial variable $x$.

Keywords: Gradient estimates, degenerate parabolic systems, nonstandard growth condition, parabolic $p$-Laplacian

Baroni Paolo, Bögelein Verena: Calderón–Zygmund estimates for parabolic $p(x, t)$-Laplacian systems. Rev. Mat. Iberoam. 30 (2014), 1355-1386. doi: 10.4171/RMI/817