Global Calderón–Zygmund estimate for p-Laplacian parabolic system

Abstract

© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.We establish a global Calderón–Zygmund theory for the weak solution of the following p-Laplacian system {ut-div(a(x,t)|∇u|p-2∇u)=-div|F|p-2F+finΩT,u=0on∂Ω×(0,T),u=u0onΩ×{0},by proving that for every q≥ p there holds |∇u0|q∈L1(Ω)and|F|q,|f|(p∗)′qp∈L1(ΩT)⟹|∇u|q∈L1(ΩT)with the desired global Calderón-Zygmund estimate, where p∗=p(n+2)n is parabolic Sobolev conjugate of p and (p∗)′ is Hölder conjugate of p∗.