regularity of weak solution to parabolic fractional p ... · pdf file1. introduction 2....

1. Introduction2. Regularity for Stationary Non Local Equations3. Regularity for Parabolic Non Local Equations
4. Future Direction
Regularity of Weak Solution to ParabolicFractional p-Laplacian
Lan Tangat BCAM Seminar
July 18th, 2012
Lan Tang at BCAM Seminar Regularity of Weak Solution to Parabolic Fractional p-Laplacian

4. Future Direction
Table of contents
1 1. Introduction1.1. Background1.2. Some Classical Results for Local Case
2 2. Regularity for Stationary Non Local Equations2.1. Linear Case2.2. Fractional p-Laplacian
3 3. Regularity for Parabolic Non Local Equations3.1. Linear Equation3.2. Parabolic Fractional p-Laplacian
4 4. Future DirectionOpen Problems

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1.1. Background1.2. Some Classical Results for Local Case
1.1. Background
Let 1 < p

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1.2. Some Classical Results for Local Case
For the pLaplacian (p > 1):
div(|u|p2u) = 0.
Uralceva (68) proved that for p > 2 , the weak solutions ofp-Laplace Equation have Holder continuous derivatives and Evans(82) gave another proof for this result.Lewis (83) and DiBenedetto (83) exteded this result to the case1 < p < 2.Uhlenbeck (77) and Tolksdoff (84) proved C 1, regularity forthe p-Laplacian systems for some > 0.Remark: In general, the weak solutions to p-Laplacian do nothave better regularity than C 1,.

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For the parabolic p-Laplacian:
ut (|u|p2u) = 0 in [a, b] .
DiBenedetto (86) proved that any weak solution is locallyHolder continuous for p > 2.For the singular case 1 < p < 2, Chen- DiBenedetto (92)proved that any essentially bounded weak solution is locallyHolder continuous.

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2.1. Linear Case2.2. Fractional p-Laplacian
In this section we consider the variational problem in non localform (p > 1 and 0 < s < 1):
min{
DD
|u(x) u(y)|p
|x y |n+psdxdy : u W s,p(D) and u = g on Rn\D}
Then the minimizer u is a fractional p-Laplacian and it satisfies theintegral equation in weak sense:
D
|u(x) u(y)|p2(u(x) u(y))|x y |n+ps
dy = 0

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2.1. Linear Case
For the simplest case p = 2, 0 < s < 1 , the equation would bereduced to
Rn
u(x) u(y)|x y |n+2s
dy = 0, u Hs
It is equivalent to ()su(x) = 0. By Caffarelli and Silvestre(2007) (An Extension Problem Related to the FractionalLaplacian, CPDE, 32 (8): 1245-1260), u can be extended tou : Rn+1 R such that u(x , 0) = u(x) and
div(|y |au) = 0 in Rn+1,
where a = 1 2s.

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In Fabes, Kenig and Serapioni (1982) ( The local regularity ofsolutions of degenerate elliptic equations. CPDE 7(1): 77-116. ),they study the degenerate elliptic equation
div(A(x)v) = 0
where cw(x)||2 TA(x) Cw(x)||2 for any Rn andw(x) > 0 is an A2-weight function i. e. there exists a uniformconstant C > 0 such that
supB
1
|B|
B
w(x)dx 1|B|
B
w(x)1dx C .
They prove that v is Holder continuous.Obviously, |y |a A2, then u C and so is u.

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For the case of measurable coefficients, i.e. the integral kernelK (x , y) satisfies : |x y |(n+2s) K (x , y) |x y |(n+2s) fortwo uniform positive constants 0 <

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Theorem A. Any bounded weak solution is locally Holdercontinuous for p > 2 and ps < p 1.The proof depends on the following oscillation lemma:Lemma 2.1. Assume that p > 2, 0 < ps < p 1 . If u is asolution to the fractional p-Laplacian and also satisfies thefollowing conditions:
|u(x)| 1 x Rn (1)|{x B1 : u(x) 0}| > > 0 (2)
where are two positive constants. Then u 1 in B1/2 forsome > 0.Remark: Proof of Lemma 2.1. is similar to that of Lemma 3.3and we will give the details there.

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3.1. Linear Equation3.2. Parabolic Fractional p-Laplacian
Now we discuss parabolic fractional p-Laplacian. Let us assumethat n 2, 0 < s < 1 and p > 1. The measurable functionK : R Rn Rn R satisfies that
K (t, x , y) = K (t, y , x) for any x 6= y
and the following : there are two positive constants and suchthat
|x y |n+ps K (t, x , y)
|x y |n+ps, x , y Rn.
Usually, the parabolic fractional p-Laplacian is like the followingform:
ut(x , t)+
Rn
K (t, x , y)|u(x)u(y)|p2(u(x)u(y))dy = 0 ()

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In the following we will introduce the weak solution of ():Definition We say w C ([a, b]; L2(Rn)) Lp([a, b]; W s,p(Rn))to be a weak solution to the parabolic fractional p-Laplacian if andonly if C0 (Rn) and t [a, b], the following holds:
RnRnK (t, x , y)|w(x) w(y)|p2(w(x) w(y))(x)dxdy
=
Rnwt(x)(x)dx .

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3.1. Linear Equation
For the special case: 0 < s < 1 and p = 2, then the equationbecomes
ut(x , t) +
Rn
K (t, x , y)(u(x) u(y))dy = 0, ()
where
|x y |n+2s K (t, x , y)
|x y |n+2sand
K (t, x , y) = K (t, y , x) for x 6= y .L. Caffarelli, C. Chan and A. Vasseur recently proved that anyweak solution to () with initial value in L2(Rn) is uniformlybounded and Holder continuous. ( see Regularity theory forparabolic nonlinear integral operators, J. Amer. Math. Soc.24(3), 849-869, 2011)

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3.2. Parabolic Fractional p-Laplacian
We assume that n 2, 0 < ps < p 1 and p > 2. Themeasurable function K : Rn Rn R is symmetric in x , y forany x 6= y and satisfies the following estimate: there are twopositive constants and such that
|x y |n+ps K (t, x , y)
|x y |n+ps, x , y Rn
Usually, the parabolic fractional p-Laplacian is like the followingform:
ut(x , t) +
Rn
K (x , y)|u(x) u(y)|p2(u(x) u(y))dy = 0.

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Theorem B. Any weak solution to the parabolic fractionalp-Laplacian ab

regularity of weak solution to parabolic fractional p ... · pdf file1. introduction 2....

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