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

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

  • Paolo Baroni

    Uppsala University, Sweden

  • Verena Bögelein

    Universität Salzburg, Austria
Calderón–Zygmund estimates for parabolic $p(x, t)$-Laplacian systems cover
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Abstract

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

∂tu−div (a(x,t)∣Du∣p(x,t)−2Du)=div (∣F∣p(x,t)−2F)\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)p(x,t). More precisely, we show that the spatial gradient DuDu of the solution is as integrable as the right-hand side FF, 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)a(x,t), requiring only a VMO condition with respect to the spatial variable xx.

Cite this article

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

DOI 10.4171/RMI/817