Regularity theory and Asymptotic behaviors in Integro-differential equations

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Regularity theory and Asymptotic behaviors in Integro-differential equations

Ki-Ahm Lee

Seoul National University www.math.snu.ac.kr/∼kiahm

Frontiers of Mathematics and Applications III - 2012, Summer Course UIMP 2012, Santander (Spain),

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1. Introduction to Nonlocal Equations

L[u](x) = (1−σ)cσ

Z

Rⁿ

hu(x+y)−u(x)−y· ∇u(x)χB₁(y)i K(y)dy where

K(y)≈ 1

|y|ⁿ⁺²σ, 0< σ <1.

Letδ(u,x,y) =u(x+y)−u(x)−y· ∇u(x)χB₁(y).

IfuisC²in a nbd ofx,

δ(u,x,y)

|y|ⁿ⁺²σ ≈|D²u(x)||y|²

|y|ⁿ⁺²σ ≈ C

|y|ⁿ⁺²σ−2

is integrable inB₁ifσ <1.

If|δ(u,x,y)| ≤C,

δ(u,x,y)

|y|ⁿ⁺²σ

< C 1+|y|ⁿ⁺²σ

for|y|>1, is integrable ifσ >0.

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Symmetry v.s. Nonsymmetry

IfK(y)is symmetric Kernels i.e.K(−y) =K(y), L[u](x) = (1−σ)c_σ

Z

Rⁿ

[u(x+y) +u(x−y)−2u(x)]K(y)dy.

The effect of “y· ∇u(x)χB₁(y)” vanishes.

(−4)^σ[u](x) = (1−σ)c_σ Z

Rⁿ

u(x+y) +u(x−y)−2u(x)

|y|ⁿ⁺²σ dy is called a fractional Laplacian.

IfKis not symmetric, then”y· ∇u(x)χB₁(y)”persists.

AndL[u](x) = (1−σ)c_σR

∂Ωδ(u,x,y)K(y)dystill has”y· ∇u(x)χB₁(y)”even thoughK(y)is symmetric since∂Ωis not symmetric with respect to the origin.

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Jump Process

The Laplacian is the infinitesimal generator of the Brownian motion which is continuous process.

L[u](x) = (1−σ)c_σ Z

Rⁿ

hu(x+y)−u(x)−y· ∇u(x)χB₁(y)i K(y)dy

describes the infinitesimal generator of a given purely jump processes,. i.e. processes without diffusion or drift part. Nonlinear integro-differential operators come from the stochastic control theory related with

Iu(x) =sup

α

L_αu(x), or game theory associated with

Iu(x) =inf

β sup

α

L_αβu(x), (1)

when the stochastic process is of L `evy type allowing jumps.

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Nonlocal Curvature

The minimizeru_ηof

η Z

|∇u|²dx+1 η

Z

F(u)dx

with a double well potential converges tou0=χΩ−χΩ^c and the interface satisfies the mean curvature equation. In phase transition ”free energy” , nonlocal energy

corresponds to processes with large scale correlation (like solidification) where information far away from the interface has direct influence in it.

E_η(u) =η Z Z

[u(x)−u(y)]²K(x,y)dxdy+1 η

Z

F(u)dx If the kernel isK ≈ ¹

|x−y|n+2σ, thenu0=χΩ−χΩ^c minimize E(Ω) = (1−σ)

Z Z

[u(x)−u(y)]²

|x−y|ⁿ⁺²σ dx or (1−σ)

Z Z χΩ(x)χΩ^c(y)

|x−y|ⁿ⁺²σ dxdy. (Other examples) Fractional Mean Curvature flows, Fractional Conformal Laplacian and fractional Yamabe problems

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2. Point of views

(1) via Fourier transformation

I (−4ˆ)^σ[u] =|ξ|²^σˆu

I (−4)^σ¹(−4)^σ²[u] = (−4)⁽^σ¹⁺^σ²⁾[u]and(−4)⁰[u] =u.

I Ifσi1, then(−4)^σ[u] =−4u.

- - - It works for linear equations, (2)

(−4)^σu=f(u) implies

u(x) = Z

Rⁿ

1

|x−y|ⁿ⁻²σf(u)dy.

Whenf(u) =uⁿ⁺²ⁿ⁻2σ^σ, thenuis invariant under Kelvin Transform and it has some scaling invariance.

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(3) Dirichlet to Neumann map

- - - Some nonlocal equation can be local equation by extending it to higher dimensional space.

Observation:4_xu^∗+u^∗_yy=0 withu^∗(x,0) =u(x)implies (−∂z)²u^∗=u^∗_yy= (−4_x)u^∗

∂yu^∗=−(−4_x)¹^/²u^∗ (2)

Aty=0,ui(−4_x)¹^/²uis a map from Dirichlet data to Neumann data.

Now we obtain the same interpretation for any fractional power(−4)^σ(0< σ <1)as the Dirichlet to Neumann operator of an appropriate extensionu^∗(x,y)satisfying the followingextension problem:











− ∇(y^a∇u^∗(x,y)) =0 inRⁿ×R⁺

u^∗(x,0) =u(x) onRⁿ

−(−4)^σu(x) =lim

y→0y^au^∗_y(x,y), (a=1−2σ), onRⁿ.

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Remark:

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(4) The Balance in the inetegral

- - - It woks for general operators. In some case, the integral can be decomposed to f(x)≥

Z

B₁

Λδ+

|y|ⁿ⁺²σ− Z

B₁

λδ+

|y|ⁿ⁺²σ − Z

Rⁿ\B₁

Λδ+−λδ+

|y|ⁿ⁺²σ

First,R

Rⁿ\B₁ Λδ+−λδ+

|y|n+2σ is bounded.

And if a concave function touchesuatxfrom above inB1, thenδ⁻=0 onB1. Now we can control

f(x) = Z

B₁

Λδ⁻

|y|ⁿ⁺²σ =X

k

Z

B2−k\B 2−(k+1)

Λδ+

|y|ⁿ⁺²σ.

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3. Harnack Inequalities and H ¨older Continuity 3.1 Elliptic Local Equation

Theorem (Harnack Inequality)

If u is a nonnegative harmonic function in B_R(0), then we have sup

B_R/2

u≤C inf

B_R/2u for a uniform constant C>0.

Corollary (Oscillation Lemma)

oscB_R/2u≤γoscB_Ru for some0< γ <1.

by applying Harnack Inequality onM(R)−uandu−m(R)forM(R) =sup_B

Ruand M(R) =infB_Ru

Theorem (H ¨older Regularity) kuk_C_α_(B

R/2)≤Ckuk_L^∞_(B

R) for some0< α <1.

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3.2 Parabolic Local Equation

Theorem (Harnack Inequality) Q_r ;Qr(0,r^σ) =Br×(0,r^σ],

Q⁻r ;Qr+ (0,r^σ) =Br×(r^σ,2r^σ)(Cube at past time) ,

Q⁺_r ;Q⁻_r + (0,2r^σ) =Br×(3r^σ,4r^σ)(Cube at future time),Qr(x0,t0) =Qr+ (x0,t0).

If u is a nonnegative caloric function inQ3R, then we have sup

Q⁻_R

u≤Cinf

Q⁺_R

u for a uniform constant C>0.

There is atime delay to control the lower bound in a small neighborhood of a point by the current value at the point, which is a main difference between elliptic and parabolic equations.

Oscillating LemmaandH ¨older Regularitycome from the similar arguments.

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Nonlocal Equations

Classical Harnack Inequality for fractional Laplacian fails ifuis nonnegative only on a bounded setBR (Kassmann). So we assumeu≥0 inRⁿ.

Harnack inequality doesn’t imply H ¨older regularity directly because of the assumption:u≥0 inRⁿ.

If the kernel is allowed to oscillate between two differentσ1andσ2, there could be a discontinuous solution (Barlow, Bass, Chen & Kassmann’09)

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Known Results in Nonlocal Equations

Probabilistic Approach

(Song& Vondracek ’04, Bass & Levin ’02, Kassmann, Chen...) Yes for Linear Equation with symmetric or nonsymmetric Kernel.

Analytic Approch (Caffarelli& Silvestre ’07)

Yes for Nonlinear Elliptic Equation with symmetric Kernel.

(Sung Hoon Kim & L. ’09)

Yes for the singular nonlocal parabolic equationut+ (−4)^σu^m=0.

(Yong-Cheol Kim & L. ’09)

Yes for Nonlinear Elliptic Equation with nonsymmetric Kernel for 1< σ <2.

Yes for Nonlinear Elliptic Equation in some large class with nonsymmetric Kernel for 0< σ <2.

(H.C. Lara and G. D ´avila ’11)

Yes for Nonlinear Elliptic Equation in some small class with nonsymmetric Kernel for 0< σ <2 after dropping off∇u.

Yes for Nonlinear Parabolic Equation with symmetric Kernel for 0< σ <2.

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Key Observations

(1.) Ellipric equation with Nonsymmetric Kernel R

Rⁿ

|y·∇u(x)χB1(y)|

|y|n+σ dyneeds to be controlled.

The equation is not scaling invariant due to|χB₁(y)|

The influence of the gradient term is very large if we want to control the solution in a small region.

The error term can be regarded as a drift term.(1< σ≤2)and(0< σ≤1) requires different technique due to the difference of the blow rate as|y| approaches to zero and the decay rate as|y|approaches to infinity.

When(1< σ≤2), better decay rate of kernel allows H ¨older regularity in a larger class where there is scaling-invariance:y· ∇u(x)χB_R(y)for allR≥1/2 has been considered.

(2.) Parabolic Equation with symmetric Kernel

The quations nonlocal in space variable , but local in time variable.

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Classes of Operators

Set

L[u](x) = (2−σ)c_σ Z

Rⁿ

hu(x+y)−u(x)−y· ∇u(x)χB₁(y)i K(y)dy,

L_αβ[u](x;∇ϕ) = (2−σ)cσ

Z

Rⁿ

hu(x+y)−u(x)−y· ∇ϕ(x)χB₁(y)i

K_αβ(y)dy, and

I[u](x;∇ϕ(x)) =sup

α inf

β L_αβ[u](x;∇ϕ).

Lemma

u is a viscosity supersolution toIu=f onΩif and only ifIu(x;∇ϕ(x))is well-defined and

Iu(x;∇ϕ(x))≤f(x)

for any x∈Ωand any C²-functionϕtouching u from below at x.

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µ(u,x,y;∇ϕ) = (u(x+y)−u(x)) +y· ∇ϕ(x)χB_R(y) µis the difference between the value ofuand its linear approximation.

µ⁺(u,x,y;∇ϕ) =max(µ(u,x,y;∇ϕ),0) µ⁻(u,x,y;∇ϕ) =max(−µ(u,x,y;∇ϕ),0).

M⁺[u](x;∇ϕ(x)) = (2−σ) Z

Rⁿ

Λµ⁺−λµ⁻

|y|ⁿ⁺σ dy, M⁻[u](x;∇ϕ(x)) = (2−σ)

Z

Rⁿ

λµ⁺−Λµ⁻

|y|ⁿ⁺σ dy.

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We consider a collection of operators such that M⁻[u](x;∇ϕ(x))≤ I[u](x;∇ϕ(x))≤ M⁺[u](x;∇ϕ(x)).

S⁺(λ,Λ,|f|)is a collection of super-solutions ofM⁻[u] =|f(x)|andS⁻(λ,Λ,|f|)is a collection of sub-solutions ofM⁺[u] =−|f(x)|.

Any solution ofI[u] =f(x)belongs toS(λ,Λ,|f|) =S⁺(λ,Λ,|f|)∩ S⁻(λ,Λ,|f|). Ifu∈ S(λ,Λ,|f|), then this class is invariant under the scaling: ^u(^η^x+x_η_σ ⁰⁾∈ S(λ,Λ,|f|) andu(ηx+x0)∈ S(λ,Λ, η^σ|f|)for 0< η <1.

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Subclass for Elliptic Nonlocal Equations with nonsymmetric kernel

Definition

Let0< η≤1andI ∈ S^L, whereLis a class of linear integro-differential operators.

Then we say thatI ∈ S^L

η if, for R∈(0,1], there areB^±

R :Rⁿ→Rsuch that

1 B^±

R(·)is homogeneous of degree one, i.e.B^±

R(0) =0andB^±

R(∇u) =B⁺

R

_∇_u

|∇u|

|∇u| for|∇u|,0,

2 |{ν∈Sⁿ⁻¹: B^±

R(ν)<0}| ≥η|Sⁿ⁻¹|>0,

3 M⁻

L,R,ηu(x)≤ Iu(x)− I0(x)≤ M⁺

L,R,ηu(x)whenever u∈C¹^,¹[x]∩B(Rⁿ)for x∈BR, whereM^±

L,R,ηu(x) :=M^±

L,Ru(x)± B^±

R(∇u(x))±(2−σ)CR¹⁻^σ|∇u(x)|. Definition

For0<R<1, the drift vectorbL,R of a linear operatorLat R is defined by bL,R= (2−σ)

Z

B₁\B_R

y K(y)dy.

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Lemma

Let0< σ <2and0<R<1. LetLbe a class of linear integro-differential operators.

Then we have the following results:

(1)IfL ∈L, thenL ∈ S^L_ηfor0< η≤ ¹

2.

(2)Assume that there is a unit vectorasuch that for any nonzero drift vectorb^L_i_,R, a,|^bb^LLⁱi,R^,R|

>0.IfI ∈ S^Lsatisfies that min

i=1,···,N

L_iu(x)≤ Iu(x)− I0(x)≤ max

i=1,···,N

L_iu(x) whenever u∈C¹^,¹[x]∩B(Rⁿ)for x∈BR, thenI ∈ S^L

ηfor someη >0.

(3)Assume that there is a vectora∈Sⁿ⁻¹andη >0such that for any nonzero drift vectorsbL_α,R,

a,|^bb^LL^α,α,^RR|

≥ηⁿ¹⁻¹ for anyα∈I. IfI ∈ S^Lsatisfies that minα∈I

L_αu(x)≤ Iu(x)− I0(x)≤max

α∈I

L_αu(x) whenever u∈C¹^,¹[x]∩B(Rⁿ)for x∈BR, then we haveI ∈ S^L_η.

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Lemma

(1)Let0< σ <2and letLbe a class of linear integro-differential operators. IfL ∈ S^L with a symmetric kernel K , thenL ∈ S^L_ηfor someη∈(0,1]. In addition, ifL_αhas symmetric kernel for allα, thensup_αL_α,inf_αL_α∈S_η^Lfor1≥η >0.

(2)If1< σ <2, thenS^L0 =S^L_η⁰ for someη∈(0,1].

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Theorems for Elliptic Nonlocal Equations

Theorem

Letσ0∈(1,2). If u is a bounded function onRⁿsuch that M⁻

L₀u≤ C₀

R^σ and M⁺

L₀u≥ −C₀

R^σ withσ0< σ <2on B2R

or

M⁻

L₀,R,ηu≤ C₀

R^σ and M⁺

L₀,R,ηu≥ −C₀

R^σ with0< σ≤1on B_2R

in the viscosity sense, then there is some constantα >0(depending only onλ,Λ,n andσ0)such that

kuk_Cα(B_R/2)≤ C R^α

kuk_L∞(Rⁿ)+C0

where C>0is some universal constant depending only onα.

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Theorem

For a givenσ0∈(0,2), letσ0< σ <2. If u∈B(Rⁿ)is a positive function such that M⁻

L₀u≤ C0

R^σ and M⁺

L₀u≥ −C0

R^σ withσ0< σ <2on B2R

or

M⁻

L₀,R,ηu≤ C0

R^σ and M⁺

L₀,R,ηu≥ −C0

R^σ with0< σ≤1on B2R

in the viscosity sense, then there is some constant C>0depending only onλ,Λ,n,σ0, andkuk_L∞(Rⁿ)such that

sup

B_R/2

u≤C inf

B_R/2u+C0

.

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Steps of Proof

Nonlocal A-B-P estimate:

It gives the lower bound of the contact set between the function and the concave envelope ,Γ(u), in terms of it’s supremum, supu⁺whenuis nonpositive outside a given domain.

Construction of BarrierΨ(x): It will be used to transform a general solutionu(x)to a functionu(x) + Ψ(x)which satisfies the configuration at the condition of A-B-P estimate.

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Decay estimate for the upper level sets

{u>t} ∩B_r(x₀)

≤Crⁿ(u(x₀) +C₀r^σ)t⁻^η⁰ . C-Z decomposition is used at this step.

Reducing the oscillation of solution with uniform factor 0< γ <1.

Main Induction: there are 0<mk <Mk <∞and 0< α <1 such that

|Mk−mk|<₁

2^k

α

andmk ≤u(x)≤Mk for allx∈B_1/2k. Idea: apply the decay estimate on

w=max u(x/2^k)−mk

(Mk+mk)/2,0

!

≥0 onRⁿ by showing thatwsatisfies a nice equation inB3/4sincew>0 inB1,

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Gradient Map

Γ(u)is the concave envelope ofuinBR.∇Γis the gradient map. SetC(u,Γ,BR)

={y∈BR:u(y) = Γ(y)}be thecontact set betweenuandΓ(u).

C⁺(u,Γ,B_R;b) =C(u,Γ,B_R)∩ {y∈B_R :b· ∇Γ(y)≥0}.B_d⊂ ∇Γ(B_R) =∇Γ(C(u,Γ,B_R)) ford=^sup_R^BR^uand

sup_B

Ru

R

!n

≤C|∇Γ|(BR). (4)

AndB_d⁺(b) ={y∈B_d :y·b>0} ⊂ ∇Γ(C⁺(u,Γ,B_R;b))is enough to have (4).

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1 < σ < 2

Lemma M⁺

Lu≥ −fon B_R, there exists some constant C>0depending only on n, λandΛ (but not onσ)such that for any x∈ C(u,Γ,BR)and any M>0there is some k∈N∪ {0} such that

Rk(x)

≤CR^σ⁻²(f(x) +J_σ(R)|∇Γ(x)|)

M |Rk(x)| (5)

whereRk(x)={y∈Rk(x) :µ⁻(u,x,y;∇Γ)≥Mr_k²}(: a bad set) andRk(x)=Br_k(x)\Br_k+1(x)for rk =%02⁻²¹⁻^σ⁻^kR,%0=1/(16√

n)(: a ring)

andJ_σ(R)is₁¹−σ(1−R¹⁻^σ)forσ∈(0,1)∪(1,2)and−log(R)forσ=1. (: a blow-up rate)

R_k(x)

measures the size of a bad set where the solution has been bended too much.

A-B-P estimate tells us that the the size of bad set is controlled byfand|∇Γ(x)|with blow-up rate. Such rate will be controlled when we prove the decay estimate of the upper level set ofufor 1< σ <2, but it blows up for 0< σ≤1.

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Idea of Proof

Γtouchesufrom above and then we haveµ(u,x,y;∇Γ)≤0 fory∈BR.

−f(x)≤ M⁺

Lu(x;∇Γ)

= (2−σ) Z

Rⁿ

−λµ⁻(u,x,y;∇Γ)

|y|ⁿ⁺σ dy+ Z

Rⁿ

Λµ⁺(u,x,y;∇Γ)

|y|ⁿ⁺σ dy

!

≤(2−σ) Z

Br0(x)

−λµ⁻(u,x,y;∇Γ)

|y|ⁿ⁺σ dy+ (2−σ)ωnΛJ_σ(R)|∇Γ(x)|

f(x0) 2−σ ≥λ

∞

X

k=0

Z

R_k(x₀)

−µ(u,x0,y;∇Γ(x))

|y|n+σ dy−ωnΛJ_σ(R)|∇Γ(x0)|

≥c

∞

X

k=0

M0

r_k²

r_k^σCR^σ⁻²f(x0) +J_σ(R)|∇Γ(x0)| M0

−ωnΛJ_σ(R)|∇Γ(x0)|.

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Thus this implies that

f(x0) + (2−σ)ωnΛJ_σ(R)|∇Γ(x0)| ≥ cρ²₀

1−2⁻⁽²⁻^σ⁾C(f(x0) +J_σ(R)|∇Γ(x0)|)

≥C(f(x0) + (2−σ)Jσ(R)|∇Γ(x0)|)

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1 The bad setRk(x)ofΓhas small measure in a ringBr_k(x)\Br_k+1(x). SinceΓis a concave function, there is no bad set in smaller ballB_r_k+1(x).

2 If we can control oscillation of the concave functionΓ, then the gradient image∇Γ will be controlled in smaller ball.

3 Then for eachxin the contact setC(u,Γ,BR), we have a rectangle where the gradient image∇Γis controllable and such rectangles will make a covering of the contact set.

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Nonlocal A-B-P estimate

Theorem

Let u andΓbe functions as in Lemma 3. Set g_η(z) = (|z|ⁿ/(n−1)+ηⁿ^/⁽ⁿ⁻¹⁾)¹⁻ⁿ. Then there exist a finite family{Qj}^m

j=1of open cubes Qjwith diameters djsuch that (a)Any two cubes Qiand Qjdo not intersect,(b)C(u,Γ,BR)⊂Sm

j=1Qj, (c)C(u,Γ,BR)∩Qj,φfor any Qj,(d)dj≤%02⁻²⁻¹σR where%0=1/(16√

n), (e)R

Q_jg_η(∇Γ(y))det(D²Γ(y))⁻dy≤CR^n(σ⁻²⁾(sup_Q

j(J_σ(R)ⁿ+η⁻ⁿ|f|ⁿ)|Qj|, (f)|{y∈4√

n Q_j:u(y)≥Γ(y)−CR⁽^σ⁻²⁾(sup_Q

j(f+J_σ(R)|∇Γ|)d²_j}| ≥η0|Q_j|, where the constants C>0andη0>0depend on n,Λandλ(but not onσ).

We may consider the nonlocal A-B-P Estimate asa kind of Riemann sumof the Local A-B-P Estimate. The sum is given with a covering of the contact set since the gradient map has contribution only on the contact set between the solution and the concave envelope. But the size of rectangle is order of 2⁻²¹⁻^σ , which goes to zero asσ→2 and then we recover the local A-B-P Estimate.

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Key Lemma

Lemma

u∈B(Rⁿ)is a viscosity supersolution toM⁻

L₀u≤ε0/R^σwithσ,1orLu≤ε0/R^σwith σ∈(σ0,2)on B₂^√_nRsuch thatu≥0onRⁿandinf_Q_Ru≤1,then |{u≤M} ∩Q_R| ≥ν|Q_R| If we apply this lemma inductively, we have the geometric decay of level sets.

Lemma

{u>t} ∩QR

≤C t⁻^ε^∗|QR|,∀t>0.

Idea of proof of Harnack InequalitySetM=sup_Q

Ruwithu(0) =1 andu≥0. We apply the geometric decay estimate onuand(M−u)₊. Then|{u≥M/2}|and

|{M−u≥M/2}|will be very small ifMis very large, but it is a contradiction since {u≥M/2} ∪ {M−u≥M/2}=QR

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First noticeC(u,Γ,BR)⊂ {u≤M} ∩QR. Apply A-B-P estimate onu+ Ψ ln

[(sup_B

2

√

nRv)/R]ⁿ

ηⁿ +1

!

≤C Z

C(u,Γ,B_R)

g_η(∇Γ(y))det[D²Γ(y)]⁻dy

≤C X

j

sup

Q_j

(R^σ⁻²J_σ(R))ⁿ+η⁻ⁿ(R⁽^σ⁻²⁾(ψ+ε0)/R^σn

|Qj|

!

≤C (R^σ⁻¹J_σ(R))ⁿ+η⁻ⁿX

j

sup

Q_j

((ψ+ε0)/R²)ⁿ|Qj|

! .

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Here we note thatKR:=Exp((R^σ⁻¹J_σ(R))ⁿ)≤C<∞for 1< σ <2 andR<1. If we setη=

P

jsup_Q

j((ψ+ε0)/R²)ⁿ|Qj|1/n

in (10), then we have that 1≤ sup

B₂^√_nR

v≤Cε0+CR X

j

(sup

Q_j

ψ)/R²n

|Qj|

!1/n

. (8)

1

21/nR≤C P

j

sup_Q

jψn

|Q_j|

!1/n

implies¹₂|Q_R| ≤C P

Q_j∩B_R/4,φ|Q_j|

!

≤ |{u≤M} ∩Q_R|.

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0 < σ ≤ 1

Lemma

For any x∈ C⁺(u,Γ,BR;b)and any M>0there is some k ∈N∪ {0}such that

eRk(x)

≤C(R^σ⁻²f(x) +R⁻¹|∇Γ(x)|)

M |Rk(x)| (9)

whereRek(x) ={y∈Rk(x) :µ⁻(u,x,y;∇Γ)≥M0r_k²}.

Take anyx∈ C⁺_η(u,Γ,B_R). From the definition ofS^L_η⁰, we have that Iu(x;∇Γ)≤ M⁺

L₀,Ru(x;∇Γ) +B⁺

R(∇Γ(x)) + (2−σ)CR¹⁻^σ|∇Γ(x)|

≤ M⁺

L₀,Ru(x;∇Γ) + (2−σ)CR¹⁻^σ|∇Γ(x)| becauseB⁺

R(∇Γ(x))≤0 from the assumption andµ⁺_R(u,x,·;∇Γ) =0 onRⁿ. The conclusion comes from similar arguments

(33)

Main Step at Key Lemma

ln [(sup_B

2

√

nRv)/R]ⁿ

ηⁿ +1

!

≤C Z

C(u,Γ,BR)

g_η(∇Γ(y))det[D²Γ(y)]⁻dy

≤C X

j

sup

Q_j

(R⁻¹J_σ(R))ⁿ+η⁻ⁿ(R⁽^σ⁻²⁾(ψ+ε0)/R^σn

|Q_j|

!

≤C (J_σ(R))ⁿ+η⁻ⁿX

j

sup

Q_j

((ψ+ε0)/R²)ⁿ|Q_j|

! .

(10)

whereJ_σ(R) =₁−¹σ(1−R¹⁻^σ)<C<∞for someC>0 since 0< σ <1.

(34)

5. Fully Nonlinear Case 5.1 Normal Map

Definition

Letu:Rⁿ×I→Rbe a function which is not positive on∂^∗pQr/2and is upper semicontinuous onQr.

(i)u(x,t)is called concave inRⁿ×Iifu(x,t)isconcave inxand nondecreasing int.

(ii)The concave envelopeΓ(y,s)ofuinQ2ris defined as Γ(y,s) =











inf{v(y,s) :v∈Π,v>u⁺inQ2r} inQ2r

0 in∂^∗pQ2r,

whereΠis the family of all concave functionsvinQ2rsuch thatv≤0 on∂pQ2r. (iii)Thenormal mapNuofu:Q→Ris given by

Nu(x,t) ={(p,h)∈Rⁿ×(τ1,t] :u(y,s)≤u(x,t) +p·(y−x),∀(y,s)∈Q,

andu(x,t)−p·x=h} (11)

(35)

Lemma

If v∈C²^,¹(Q_r(x₀,t₀))is concave in x and increasing in t and if v=0on∂pQ_r(x₀,t₀), then we have that

(a) Z

B_r(x₀)

v(x,t0)det(D²v(x,t0))dx

= (n+1) Z

Q_r(x₀,t₀)

∂tv(x,t)det(D²v(x,t))dxdt, (b)

Nv(Qr) = Z

Qr

∂tv(x,t)det(D²v(x,t))⁻dxdt, (c)

max

x∈B_r v(x,t0)≤Cr (Nv(Qr(x0,t0)))ⁿ⁺¹¹ .

(36)

5.2 Main Steps at Parabolic A-B-P Estimate

1 Choose any point in the contact setC(u,Γ,Q1) :={(x,t) : u(x,t)Γ(x,t)}. Note date det(D²u(x,t)) =0 inQ1\C(u,Γ,Q1).

2 For a fixed time, carry out elliptic argument to control bad set in a cube of space variable .

3 Control the growth rate concave envelope at a cube of space-time variable using the property of the concave envelope.

For any(x,t)∈Q1∩ C, we have that 0≤sup

B_r(x)

Γt(y,t)≤ (1−cr) sup

B_r(x,t)∩C

Γt(y,t) +1 crsup

Q₁

Γt(y,s)

! onQ1

for some uniform constantc>0.

(37)

Parabolic A-B-P Estimate

Theorem

There exist a finite family{Kk}of pairwise disjoint(n+1)-dimensional cubes with sidelength dk such that

(a) C(u,Γ,Q₁)⊂S

kK_k,

(b) C(u,Γ,Q1)∩Kk ,φfor any k , (c) rk ≤2ρ02⁻²¹⁻σ for any k , (d) |NΓ(K_k)| ≤C

sup_2K

kf+d_ksup_Q

1fn+1

× |C(u,Γ,Q₁)∩K_k| (e)

sup_Q₁u⁺n+1

≤C|NΓ(Q1)|

where the constants C>0depends on n,Λandλ(but not onσ)and2Kk is cube with 2d_k containing K_k.

(38)

6. Degenerate nonlocal Equation of Porous Medium Type (Joint work with Sunghoon Kim)

In this chapter, 0< σ <1.σ=1 corresponds to the standard Laplacian. We consider initial value problem with fractional fast diffusion:











(−4)^σu^m+ut=0 inΩ

u=0 onRⁿ\Ω

u(x,0) =u0(x) non-negative andH˙₀^σ-bounded

(12)

in the range of exponents ⁿ_n+2σ⁻^2σ <m<1, with 0< σ <1.











(−4)^σv+ v^m¹

t =0 inΩ

v=0 onRⁿ\Ω

v(x,0) =v0(x) =u^m₀(x) inΩ,

(M.P)

(39)

We consider the extension problem.











∇(y^a∇v^∗) =0 iny>0 lim

y→0y^av_y^∗(x,y,t) =(v^m¹)t(x,0,t) x∈Ω v^∗(x,0,t) =0 onRⁿ\Ω

(13)

fora=1−2σ. Since the diffusion coefficientsD(v) =|v|¹⁻m¹ goes to infinity asv→0, we need to control the oscillation ofvfrom below. Hence, we consider the new function w^∗derived fromv^∗such thatw^∗(x,y,t) =M−v^∗(x,y,t+t₀)with

M=M(t0) =sup_t≥t₀>0v^∗. After showing the bound ofL^∞-norm ofv, we know that the solution satisfies

v^∗(·,t)≤M(t₀)<∞ (t≥t₀).

From this, we get to a familiar situation:











∇(y^a∇w^∗) =0 iny>0

− lim

y→0⁺y^a∇_yw^∗(x,y) =h

(M−w^∗)^m¹i

t(x,0) x∈Ω

w^∗(x,0,t) =M onRⁿ\Ω.

(14)

(40)

Theorem

(From L^n+2σⁿ⁻^2σ to L^∞) Let v(x,t)be a function in L^∞(0,T;Lⁿ²ⁿ⁻2σ(Ω))∩L²(0,T; ˙H^σ₀(Rⁿ)), then

sup

x∈Ω

|v(x,T)| ≤C^∗ kv0k

Ln²ⁿ−2σ(Ω)

T^2mn⁻⁽ⁿ⁻^mn²^σ)(1+m) for some constant C^∗>0.

For the second theorem, we need better control ofv.

Theorem (H ¨older regularity of fractional FDE)

For x0= (x₀¹,· · ·,x₀ⁿ), we define Qr(x0,t0) = [x₀ⁱ −r,x₀ⁱ +r]ⁿ×[t0−r²^σ,t0], for t0>r^2σ>0. Assume now that[x₀ⁱ −r,x₀ⁱ+r]ⁿ⊂Ωand v(x,t)is bounded in

Rⁿ×[t0−r²^σ,t0], then there exist constantsγandβin(0,1)that can be determined a priori only in terms of the data, such that v is C^βin Q_γr(x0,t0).

(41)

Properties

1 (Scaling Invariance)

u⁰(x⁰,t⁰) =L^m^2σ⁻¹T⁻^m¹⁻¹u x⁰

L,t⁰ T

2 (L¹-contraction) Z

Ω

hu(x,t)−˜u(x,t)i

+dx≤ Z

Ω

hu(x, τ)−˜u(x, τ)i

+dx (15)

As a consequence,

ku(t)−u(t)˜ k₁≤ ku0−˜u0k₁. (16)

3 Comparison Principle

4 Mass conservation in Cauchy problem.R

Rⁿu0(x)dx=R

Rⁿu(x,t)dxfort>0.

5 (Extinction in Finite Time in the bounded domain with zero data) There exists T^∗>0 such thatu(·,t) =0 for allt≥T^∗, i.e.,

lim

t→T^∗

ku(·,t)k∞=0

6 (No waiting time) Even foru0(x)having a compact support,u(x,t)>0 fort>0.

(42)

Estimates on Finite Extinction Time

Lemma

When_n+2ⁿ⁻²^σ_σ <m<1, there exists a positive constant C such that the solution v=u^m of (M.P)satisfies

T^∗−t≤C Z

Rⁿ

v^m+1^m (x,t)dx

!_1+m¹⁻^m .

Lemma

When_n+2ⁿ⁻²^σ_σ <m<1, the solution v satisfies Z

Rⁿ

v^m+1^m (x,t)dx≤

1− t T^∗

^1+m1−mZ

Rⁿ

v^m+1^m (x,0)dx. (17)

(43)

Proof of Theorem for L

^∞

-norm

Multiplying the equation by the functionvk = (v−Ck)₊and integrating in space,Rⁿ, we have

1 m

Z

Rⁿ

d dt

"Z v_k

0

(ξ+Ck)^m¹⁻¹ξdξ

# dx+

Z

Rⁿ

vk[(−4)^σvk]dx≤0 (18) since

(Ck +vk)^m¹vk = (Ck+ξ)^m¹ξ

ξ=v_k ξ=0 =

Z v_k

0

d dξ

h(Ck+ξ)^m¹ξi dξ.

After some computation, we haveUk ≤2C(m+1) R

Ωv_k^1+m^m (s)dx _1+m^2m

for Uk =sup

t≥T_k

Z

Ω

v

1 m+1

k dx+ (m+1)

Z ^∞

T_k

kvkk²

H˙σdt. By applying time average and H ¨older regularity,

U_k ≤C⁰ 2^1+m^4m

1+^(1+m)σ_mn

− 2 1+m

!k

t¹⁻

1−m 1+m 0 N^1+m^4m

1+^(1+m)σ_mn

−2

U

2m 1+m

1+^(1+m)σ_mn

k−1

(44)

Proof of Theorem for C

^α

-norm

1 For the technical reason, we consider the equation for thew=maximum−v.

2 Wr consider local Energy estimate for theextension problemsince the values on the local region has large influence from the whole domain. Technically the multiple of the cut-off functionηandwis hard to applied the nonlocal operator.

And the extension problem is easy to handle since it is a local equation. But the cut-off functionis only depends onx, notz. Still it can be played as an cut-off function in(x,z)variable for a little bit larger level set. And we need to useintrinsic scale, for example the scaled parabolic cylinder is

Q_k(θ0) =B_R_k×

−θ⁻₀^αR_k²^σ,0

3 If localL²-norm is uniformly small forusuch that|u|<1 in a parabolic cylinder,

|u|<1/2 in a half cylinder as the control forL^∞-norm.

4 For general case, there is a timet^∗, when the solution is close to the supremum with nontrivial measure. Through DiBenedetto’s trick, we can show the solution is close to the supremum with nontrivial measure fort^∗<t<0.

5 Use De Giorgi’s isoperimetric inequality to show that the upper level set is

uniformly small after uniform step. Then the localL^∞-estimate says the supremum decreases with a uniform amount.

(45)

Application to the asymptotic behavior for the parabolic flows

Theorem

Under the above assumptions on u₀and m, we have the following property near the extinction time of a solution u(x,t): for any sequence{u(x,tn)}, we have a

subsequence tn_k →T^∗and aϕ(x)such that lim

k→∞(T^∗−t_n_k)⁻¹^/⁽¹⁻^m)

u(x,t_n_k)−U(x,t_n_k;T^∗) →0

uniformly in compact subset ofΩfor U(x,t;T^∗) = (T^∗−t)¹^/⁽¹⁻^m)ϕ¹^/^m(x)whereϕis a eigen-function of fully nonlinear equation











(−4)^σϕ= 1

1−mϕ^m¹ inΩ

ϕ=0 onRⁿ\Ω

ϕ >0 inΩ.

(46)

7.Geometry of the Ground State (Joint work with Sunghoon Kim)











(−4)¹²ϕ=λϕ^p inΩ

ϕ >0 inΩ

ϕ=0 onRⁿ\Ω.

(19) The main question we address is motivated by the following conjecture:

Conjecture

Letϕσbe the ground state eigenfunction for the symmetric stable processes of index 0< σ <1killed upon leaving the interval I= (−1,1). Thenϕ_σis concave on I.

Rodrigo Ba ñuelos, Tadeusz Kulczycki, Pedro J. M éndez-Hern ández.On the shape of the ground state eigenfunction for stable processes.Potential Anal. 24(2006), no. 3

(47)

The solutionuof











(−4)^σu+ut =0 (x,t)∈Rⁿ×[0,∞) u(x,t)>0 x∈Rⁿ

u(x,t) =0 x∈Rⁿ\Ω u(x,0) =g(x) x∈Rⁿ

is the trace,u^∗(x,0,t), ofu^∗satisfying the followingextension problem:











4_xu^∗+u^∗_,_zz=0 inRⁿ×R⁺×[0,∞) u^∗_z(x,0,t) =u^∗_t(x,0,t) x∈Ω u^∗(x,0,t) =0 x∈Rⁿ\Ω u^∗(x,0,0) =g(x) x∈Rⁿ.

This relation betweenuandu^∗is very useful to discuss the geometric properties.

(48)

Theorem

LetΩbe a convex bounded domain, and let u0≥0be a continuous and bounded initial function. If(u0)⁻ⁿ⁺¹² is strictly convex, then the solution u^∗is power-convex in the space variable x for all t,z>0, i.e., D_x²(u^∗)⁻ⁿ⁺¹² ≥0.

Corollary

Under the same condition, the stationary profileϕ(x)of u(x,t)is power-convex, i.e., D²_x(ϕ(x))⁻ⁿ⁺¹² ≥0.

Lemma

IfΩis smooth and strictly convex, thenϕ(x)is strictly power-convex: there exists a constant c1>0such that

D_x²(ϕ(x))⁻ⁿ⁺¹² ≥c₁I. The constant c1depends onϕandΩ.

(49)

Parabolic Approximation and Nontriviality of the Limit

Lemma (Approximation Lemma) For every u0∈H˙

1 2

0(Ω), we have

|e^λ¹^tu(x,t)−a1ϕ(x)| ≤Ce⁻⁽^λ²⁻^λ¹^)t (20) and

ke^λ¹^tu(x,t)−a1ϕ(x)k

C^k_x(Ω)≤CKe⁻⁽^λ²⁻^λ¹^)t (21) for k=1,2,· · ·.

Put

e^λ¹^tu(x,t)−a1ϕ(x) =e⁻^(λ²⁻^λ¹^)tη(x,t).

Then,η(x,t)satisfies the equationηt+ (−4)¹²η=λ2η. The regularity onηgives the conclusion.

(50)

Global Regularity and Interior Regularity

Lemma

If u^∗is a solution of the problem, then the function u^∗is bounded and ku^∗k_L∞(Rⁿ×R⁺)≤sup

x∈Rⁿ

u0(x). ku^∗_tk_L∞

(Rⁿ×R⁺)≤Cku0k_H_1/2_(R_n₎ ku^∗_zk_L^∞_(Ω)≤C at z=0

(51)

Lemma

If u^∗is a solution of the problem , and if|u^∗| ≤1on the cylinder B₁⁺×[0,1], Then

|u^∗_x_i| ≤C, |u_x^∗_i_x_i| ≤C

|u^∗_zz| ≤C (i=1,· · ·,n)on the cylinder B₁⁺_/₂×h₁

2,1i

for some constant C>0.

(i) (Bernstein Computation) Find a sub-solution to apply maximum principle:

X=A u^∗²+η²(u^∗_x_i)² where A is a large number andηis a function satisfying

η(x,z,t) = e^z

2t

2 + t

2 z+t² 4

!

−1

!

(1− |x|²−z²). Observe(u_x^∗

i)²≤XonB₁⁺_/₂×h

1 2,1i

. (ii) Apply the same method for

X =A u^∗_x

i

2+η²(u^∗_x

i,xi)²

(52)

Boundary Regularity

Lemma (Boundary Regularity foru)

LetΩbe a convex subset ofRⁿwith n>1and u is the solution. Then,

u²(x,t)−c0dist(x, ∂Ω)

≤C0(dist(x, ∂Ω))¹⁺^γ, t>0 near the boundary∂Ωfor some uniform constant C₀and0< γ <1.

First we have

c1(dist(x, ∂Ω))¹² ≤u(x,t)≤c2(dist(x, ∂Ω))¹², near the boundary∂Ω. (22) ustays away from the upper bound or lower bound at an interior point with a fixed amountδ. Harnack estimate in the interior gives a ball whereustays away from the upper or lower bound. Such improvement in a ball with uniform measure will make improvement all the way to the boundary by comparing super- or sub-solution having the improvement. Then we have improvement.

c3

dist x, ∂Ω1

4

¹₂

≤u1

4(x,t)≤c4

dist x, ∂Ω1

4

¹₂

, 3t1

4 ≤t≤t1

for some constantsc3<c4such that

Regularity theory and Asymptotic behaviors in Integro-differential equations