Gradient regularity for nonlinear parabolic equations
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Tuomo Kuusi [1] ; Giuseppe Mingione [2]
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[1] Aalto University
Aalto University
Helsinki, Finlandia
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[2] Università di Parma, Italia
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- Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 12, Nº 4, 2013, págs. 755-822
- Idioma: inglés
- Enlaces
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Resumen
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.
