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

Paolo Baroni; Verena Bögelein

doi:10.4171/rmi/817

JournalsrmiVol. 30, No. 4pp. 1355–1386

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

Paolo Baroni
Uppsala University, Sweden
Verena Bögelein
Universität Salzburg, Austria

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Abstract

We prove local Calderón–Zygmund estimates for weak solutions of the evolutionary $p (x, t)$ -Laplacian system

\partial_{t} u - div (a (x, t) ∣ D u ∣^{p (x, t) - 2} D u) = div (∣ F ∣^{p (x, t) - 2} F)

under the classical hypothesis of logarithmic continuity for the variable exponent $p (x, t)$ . More precisely, we show that the spatial gradient $D u$ of the solution is as integrable as the right-hand side $F$ , i.e.,

∣ F ∣^{p (\cdot)} \in L_{loc}^{q} ⟹ ∣ D u ∣^{p (\cdot)} \in L_{loc}^{q} 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 $x$ .

Cite this article

Paolo Baroni, Verena Bögelein, Calderón–Zygmund estimates for parabolic $p (x, t)$ -Laplacian systems. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1355–1386

DOI 10.4171/RMI/817