Resumen de Gradient regularity for nonlinear parabolic equations

Tuomo Kuusi, Giuseppe Mingione

• We consider non-homogeneous degenerate and singular parabolic equations of the p-Laplacian type and prove pointwise bounds for the spatial gradient of solutions in terms of intrinsic parabolic potentials of the given datum.

  In particular, the main estimate found reproduces in a sharp way the behavior of the Barenblatt (fundamental) solution when applied to the basic model case of the evolutionary p-Laplacian equation with Dirac datum. Using these results as a starting point, we then give sufficient conditions to ensure that the gradient is continuous in terms of potentials; in turn these imply borderline cases of known parabolic results and the validity of well-known elliptic results whose extension to the parabolic case remained an open issue. As an intermediate result we prove the H¨older continuity of the gradient of solutions to possibly degenerate, homogeneous and quasilinear parabolic equations defined by general operators.