Calderón–Zygmund estimates for parabolic $p(x, t)$-Laplacian systems
Paolo Baroni, Verena Bögelein · Revista Matemática Iberoamericana · 2014
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\text{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 .