We consider the evolutionary (Formula Presented)-Laplacean system (Formula Presented), and prove the continuity of the spatial gradient (Formula Presented) under the Lorentz space assumption (Formula Presented) is time independent the condition improves in (Formula Presented). This is the limiting case of a result of DiBenedetto claiming that (Formula Presented) is Hölder continuous when (Formula Presented) for (Formula Presented). At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system (Formula Presented) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions.
Borderline gradient continuity for nonlinear parabolic systems / Kuusi, Tuomo; Mingione, Giuseppe. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - 360:3-4(2014), pp. 937-993. [10.1007/s00208-014-1055-1]
Borderline gradient continuity for nonlinear parabolic systems
MINGIONE, Giuseppe
2014-01-01
Abstract
We consider the evolutionary (Formula Presented)-Laplacean system (Formula Presented), and prove the continuity of the spatial gradient (Formula Presented) under the Lorentz space assumption (Formula Presented) is time independent the condition improves in (Formula Presented). This is the limiting case of a result of DiBenedetto claiming that (Formula Presented) is Hölder continuous when (Formula Presented) for (Formula Presented). At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system (Formula Presented) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
