Borderline Gradient Continuity for the Normalized p-Parabolic Operator
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 borde...
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Published in | The Journal of geometric analysis Vol. 33; no. 8 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
01.08.2023
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 1050-6926 1559-002X |
DOI: | 10.1007/s12220-023-01317-7 |