Borderline Gradient Continuity for the Normalized p-Parabolic Operator

• Published in: The Journal of geometric analysis Vol. 33; no. 8
• Main Authors: Akman, Murat, Banerjee, Agnid, Munive, Isidro H.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2023; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Continuity (mathematics); Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Estimates; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Theorems; Primary 35K55; Viscosity solutions to PDEs; 35B65; 35D40; Normalized p-Poission equation; Borderline regularity of solutions to PDEs; Riesz Potential
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01317-7

In this paper, we prove gradient continuity estimates for viscosity solutions to Δ p N u - u t = f in terms of the scaling critical L ( n + 2 , 1 ) norm of f , where Δ p N 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 P n + 1 f as defined in ( 2.9 ) below. Moreover, for f ∈ L m 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 Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) which considers bounded f . Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in Banerjee and Munive (Commun Contemp Math 22(8):1950069, 2020). Moreover differently from that in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018), our approach is independent of the Ishii–Lions method which is crucially used in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.