Akman, Murat and Banerjee, Agnid and Munive, Isidro H (2023) Borderline gradient continuity for the normalized p-parabolic operator. Journal of Geometric Analysis, 33 (8). DOI https://doi.org/10.1007/s12220-023-01317-7
Akman, Murat and Banerjee, Agnid and Munive, Isidro H (2023) Borderline gradient continuity for the normalized p-parabolic operator. Journal of Geometric Analysis, 33 (8). DOI https://doi.org/10.1007/s12220-023-01317-7
Akman, Murat and Banerjee, Agnid and Munive, Isidro H (2023) Borderline gradient continuity for the normalized p-parabolic operator. Journal of Geometric Analysis, 33 (8). DOI https://doi.org/10.1007/s12220-023-01317-7
Abstract
In this paper, we prove gradient continuity estimates for viscosity solutions to ΔNpu−ut=f in terms of the scaling critical L(n+2,1) norm of f, where ΔNp is the game theoretic normalized p−Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for u in terms of the modified parabolic Riesz potential Pfn+1 as defined in (2.8) below. Moreover, for f∈Lm with m>n+2, we also obtain Hölder continuity of the spatial gradient of the solution u, see Theorem 2.6 below. This improves the gradient Hölder continuity result in [3] which considers bounded f. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in [9]. Moreover differently from that in [3], our approach is independent of the Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.
Item Type: | Article |
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Uncontrolled Keywords: | Normalized p-Poission equation; Borderline regularity of solutions to PDEs; Viscosity solutions to PDEs; Riesz Potential |
Divisions: | Faculty of Science and Health Faculty of Science and Health > Mathematics, Statistics and Actuarial Science, School of |
SWORD Depositor: | Unnamed user with email elements@essex.ac.uk |
Depositing User: | Unnamed user with email elements@essex.ac.uk |
Date Deposited: | 28 Jun 2023 09:29 |
Last Modified: | 16 May 2024 21:36 |
URI: | http://repository.essex.ac.uk/id/eprint/34167 |
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