Abstract
Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.
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Acknowledgements
This research was initiated during a visit of the first-named author at the Institut Mittag-Leffler in August 2017. He thanks the Director and the Staff of the Institut for their support and hospitality. The authors wish to thank the referees for their valuable comments and suggestions.
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This research was partly funded by the Research Project of the Italian Ministry of University and Research (MIUR) Prin 2012 “Elliptic and parabolic partial differential equations: geometric aspects, related inequalities, and applications” (Grant No. 2012TC7588), GNAMPA of the Italian INdAM—National Institute of High Mathematics, and RUDN University Program 5-100.
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Cianchi, A., Maz’ya, V.G. Second-Order Regularity for Parabolic p-Laplace Problems. J Geom Anal 30, 1565–1583 (2020). https://doi.org/10.1007/s12220-019-00213-3
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DOI: https://doi.org/10.1007/s12220-019-00213-3
Keywords
- Nonlinear parabolic equations
- Nonlinear parabolic systems
- Second-order derivatives
- p-Laplacian
- Cauchy–Dirichlet problems
- Convex domains