JournalsrmiVol. 28, No. 2pp. 535–576

Potential estimates and gradient boundedness for nonlinear parabolic systems

  • Tuomo Kuusi

    Aalto University, Finland
  • Giuseppe Mingione

    Università di Parma, Italy
Potential estimates and gradient boundedness for nonlinear parabolic systems cover
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Abstract

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary pp-Laplacean system

utdiv(Dup2Du)=V(x,t),u_t - \operatorname{div} (|Du|^{p-2}Du)=V(x,t) ,

and provide LL^\infty-bounds for the spatial gradient of solutions DuDu via nonlinear potentials of the right hand side datum VV. Such estimates are related to those obtained by Kilpeläinen and Malý [22] in the elliptic case. In turn, the potential estimates found imply optimal conditions for the boundedness of DuDu in terms of borderline rearrangement invariant function spaces of Lorentz type. In particular, we prove that if VL(n+2,1)V\in L(n+2,1) then DuLlocDu \in L^\infty_{\mathrm{loc}}, where nn is the space dimension, and this gives the borderline case of a result of DiBenedetto [5]; a significant point is that the condition VL(n+2,1)V \in L(n+2,1) is independent of pp. Moreover, we find explicit forms of local a priori estimates extending those from [5] valid for the homogeneous case V0V \equiv 0.

Cite this article

Tuomo Kuusi, Giuseppe Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 535–576

DOI 10.4171/RMI/684