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Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality | | Hongjie Dong
; Peng Fa
; Yi Ru-Ya Zhang
; Yuan Zhou
; | | Date: |
5 Aug 2019 | | Abstract: | Denote by $Delta$ the Laplacian and by $Delta_infty $ the
$infty$-Laplacian. A fundamental inequality is proved for the algebraic
structure of $Delta vDelta_infty v$: for every $vin C^infty$, $$left| {
|D^2vDv|^2} - {Delta v Delta_infty v } -frac12[|D^2v|^2-(Delta
v)^2]|Dv|^2
ight| le frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on
this, we prove the following results:
1. For any $p$-harmonic functions $u$, $pin(1,2)cup(2,infty)$, we have
$$|Du|^{frac{p-gamma}2}Duin W^{1,2}_{
m loc},$$
with $gamma<min{p+frac{n-1}{n},3+frac{p-1}{n-1}}$. As a by-product,
when $pin(1,2)cup(2,3+frac2{n-2})$, we reprove the known $W^{2,q}_{
m
loc}$-regularity of $p$-harmonic functions for some $q>2$.
2. When $nge 2$ and $pin(1,2)cup(2,3+frac2{n-2})$, the viscosity
solutions to parabolic normalized $p $-Laplace equation have the $W_{
m
loc}^{2,q}$-regularity in the spatial variable and the $W_{
m
loc}^{1,q}$-regularity in the time variable for some $q>2$. Especially, when
$n=2$ an open question in [17] is completely answered.
3. When $nge 1 $ and $pin(1,2)cup(2,3)$, the weak/viscosity solutions to
parabolic $p $-Laplace equation have the $W_{
m loc}^{2,2}$-regularity in the
spatial variable and the $W_{
m loc}^{1,2}$-regularity in the time variable.
The range of $p$ (including $p=2$ from the classical result) here is sharp for
the $W_{
m loc}^{2,2}$-regularity. | | Source: | arXiv, 1908.1547 | | Services: | Forum | Review | PDF | Favorites | |
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