数理解析研究所講究録
第 1779 巻 2012 年 69-87

69

REGULARITY THEORY AND ASYMPTOTIC BEHAVIORS IN
THE INTEGRO-DIFFERENTIAL EQUATIONSSUNGHOON KIM, YONG-CHEOL KIM, AND KI-AHM LEE

ABSTRACT. In this paper, we are going to discuss the recent development in the nonlocal the integro-differential equations and theirapplications.

1. INTRODUCTION
In this paper, we are going to consider nonlocal equations. The
linear elliptic and parabolic integro-dffferential operators are given as(1)$\mathcal{L}u(x)=p.v.\int_{\mathbb{R}^{n}}\mu(u,x,y)K(y)dy$

for $\mu(u,x,y)=u(x+y)-u(x)-(\nabla u(x)\cdot y)\chi_{B_{1}}(y)$, which describes theinfimitesimal generator of given purelyjump processes, i.e. processes
without diffusion or drift part [CS]. We refer the detailed definitionsof notations to [CS, KLl, KL2]. Then we see thatis well-definedprovided that
wheredenotes the family of
all real-valued bounded functions defined on
and
means
function at . If is symmetric $(i.e. K(-y)=K(y))$ , then an oddfunction
will be canceled in the integral, andso we have that$\mathcal{L}u(x)$$u\in C_{\chi}^{1,1}(x)\cap B(\mathbb{R}^{n})$$B(\mathbb{R}^{n})$$C_{\chi}^{1,1}(x)$$\mathbb{R}^{n}$$x$$C^{1,1_{-}}$$K$$[(\nabla u(x)\cdot y)\chi_{B_{1}}(y)]K(y)$$\mathcal{L}u(x)=$p.v. $f_{\mathbb{R}^{n}}[u(x+y)+u(x-y)-2u(x)]K(y)dy$.

Nonlinear integro-differential operators come from the stochastic
control theory related with$\mathcal{I}u(x)=\sup_{\alpha}$

L. $u(x)$ ,or game theory associated with(2)$\mathcal{I}u(x)=\inf_{\beta}\sup_{\alpha}\mathcal{L}_{\alpha\beta}u(x)$,when the stochastic process is of L\‘evy type allowing jumps; see[ , CS, KLl]. Also an operator like tu $(x)= \sup_{\alpha}$ in
can be$S$$f_{\beta}\mathcal{L}_{\alpha\beta}u(x)$70
SUNGHOON KIM, YONG-CHEOL KIM, AND KI-AHM LEE

considered. Characteristic properties of these operators can easily bederived as follows;.
(3)$\inf l_{\alpha\beta}v(x)\leq I[u+v](x)-tu(x)\leq\sup \mathcal{L}_{a\beta}v(x)$$a\beta$$\alpha\beta$

1.1. Operators. In this section, we introduce a class of operators.
All notations and the concepts of viscosity solution follow [CS] withminor changes.
For parabolic setting and our purpose, we shall consider functions$u(x)$ defined on
and restrict our attention to the operators
where the measure is given by a positive kemel which is symmetric.
That is to say, the operators are given by$\mathcal{L}$$\mathbb{R}^{n}\cross[0,T]$$K$$\mathcal{L}$$lu(x)=$ p.v. $\int_{\mathbb{R}^{n}}\mu(u,x,y)K(y)dy$(4)

where $p(u,x,y)=u(x+y)+u(x-y)-2u(x)$ . And we consider theclass 2 of the operators associated with positive kemels $K\in K_{0}$satisfying that$\mathcal{L}$(5)$(2- \sigma)\frac{\lambda}{|y|^{n+\sigma}}\leq K(y)\leq(2-\sigma)\frac{\Lambda}{|y|^{n+\sigma}}\prime 0<\sigma<2$.

The maximal operator and the minimal operator with respect to 2
are defined by(6)$\Lambda t_{\mathfrak{L}}^{+}u(x)=\sup_{l\in \mathfrak{L}}Lu(x)$and$\mathcal{M}_{\mathfrak{L}}^{-}u(x)=l\in \mathfrak{L}jMLu(x)$.

be a bounded open domain. For
In what follows, we let$(x)\in\Omega$ and a function:
which is semicontinuous on ), we say) and(resp.
that belongs to the function class
if there exists asuch
we writeresp.
for some openthat $\varphi(x)=u(x)$ and $\varphi>u$ (resp. $\varphi<u$ ) onof , where . We note that geometricallyneighborhoodandhaving a local maximum at $(x)$ in is equivalent tohaving a local minimmum at $(x)$ in is equivalent toand $Iu(x)$ may be written as
And the expression for$\Omega\subset \mathbb{R}^{n}$$u$$\mathbb{R}^{n}$$C_{\Omega}^{2}(u,x)^{-}$$C_{\Omega}^{2}(u,x)^{+}$$\varphi$$\varphi\in C_{\Omega}^{2}(u,\cdot x)^{+}$$\varphi\in C_{\Omega}^{2}(u,x)^{-})$$($$U_{t,\delta}$$U\backslash \{(x)\}$$U\subset\Omega$$\chi$$U$$u-\varphi$$\varphi\in C_{\Omega}^{2}(u,x)^{+}$$\Omega$$\varphi\in C_{\Omega}^{2}(u,x)^{-}$$\Omega$$u-\varphi$$\mathcal{L}_{a\beta}u(x)$,$\mathcal{L}_{a\beta}u(x)=\int_{\mathbb{R}^{n}}\mu(u,x,y)K_{a\beta}(y)dy$tu$(x)= iffl\sup_{\alpha}\mathcal{L}_{\alpha\beta}u(x)\beta$’

71$NomoCAL$ EQUATIONS

where
and

$K_{\alpha\beta}\in K_{0}$$\mathcal{M}_{\mathfrak{L}}^{-}u(x)$. Then we see$M_{\mathfrak{L}}^{-}u(x)\leq tu(x)\leq \mathcal{M}_{\mathfrak{L}}^{+}u(x)$have the following simple forms,, and$\mathcal{M}_{\mathfrak{L}}^{+}u(x)=(2-\sigma)f_{\mathbb{R}^{n}}\frac{\Lambda\mu^{+}(u,x,y)-\lambda\mu^{-}(u,x,y)}{|y|^{n+\sigma}}dy$,$\mathcal{M}_{\mathfrak{L}}^{-}u(x)=(2-\sigma)f_{\mathbb{R}^{n}}\frac{\lambda\mu^{+}(u,x,y)-\Lambda\mu^{-}(u,x,y)}{|y|^{n+\sigma}}dy$,$\mathcal{M}_{\mathfrak{L}}^{+}u(x)$

(7)

where

$\mu^{+}$

and$\mu^{-}$are given by$\mu^{\pm}(u,x, y)=\max\{\pm\mu(u, x, y), 0\}$.

A function :is said to be(we writeat, if there are $r_{0}>0$ and $M>0$ (independent of s) such that$u$$\mathbb{R}^{n}arrow \mathbb{R}$$C_{x,\pm}^{1,1}$$(x)\in \mathbb{R}^{n}$$u\in C_{x,\pm}^{1,1}(x))$(8)$\pm(u(x+y)+u(x-y)-2u(x))\leq M|y|^{2}$for any.
We write
if
for any $(x)\in U$ and the constant$M$ in (8) is$(x)$
independent of , where
for some $\delta>0$ for an
open subset of . And we denote$y\in B_{r_{0}}(0)$$u\in C_{x,\pm}^{1,1}(U)$$u\in C_{x}^{1}|_{\pm}^{1}(x)$$U\subset \mathbb{R}^{n}$$U$$C_{x}^{1,1}(x)=C_{x,+}^{1,1}(x)\cap C_{\chi,-}^{1,1}(x),and$$\mathbb{R}^{n}$$C_{x}^{1,1}(U)=C_{x,+}^{1,1}(U)\cap C_{x,-}^{1,1}(U),$.

We note that if, then tu $(x)$ andwill be welldefined. We shall use these maximal and minimal operators to obtainregularity estimates.
Let
where ’s are all the kernels of all operators
in a class . For any class , we shall assume that$u\in C_{x}^{1,1}(x)$$K(x)= \sup_{\alpha}K_{\alpha}(x)$$\mathcal{M}_{\mathfrak{L}}^{\pm}u(x)$$K_{\alpha}$$\mathfrak{L}$$\mathfrak{L}$(9)$f_{\mathbb{R}^{n}}(|y|^{2}\wedge 1)K(y)dy<\infty$.

The following is a kin of operators of which the regularity result
shall be obtained in this paper.
$d$Definition 1.1. Let be a class of linear integro-dffferential operators.
Assume that (9) holds for . Then we say that an operator $J$ is elliptic withrespect to 8, if it satisfies the following properties:$(a)Ju(x)$ is well-defined for any.
$(b)Ju$ is continuous on an open
, whenever.
, then we have that
If$\mathfrak{L}$$\mathfrak{L}$$u\in C_{x}^{1,1}(x)\cap B(\mathbb{R}^{n})$$\Omega\subset \mathbb{R}^{n}$

$(c)$

(10)

$u\in C_{x}^{1,1}(\Omega)\cap B(\mathbb{R}^{n})$$u,v\in C_{x}^{1,1}(x)\cap B(\mathbb{R}^{n})$$\Lambda t_{\mathfrak{L}}^{-}[u-v](x)\leq Ju(x)-Jv(x)\leq \mathcal{M}_{\mathfrak{L}}^{+}[u-v](x)$.

And We denote by
the class of integro-dffferential opemtors which is
elliptic with respect to 2.
$S^{\mathfrak{L}}$72
SUNGHOON KIM, YONG-CHEOL KIM, AND KI-AHM LEE

1.2. Fourier Transformation and Dirichlet to Neumann Map. In
this subsection, we are going to discuss fractional Laplacian intermsof Fourier transformation and Dirichlet to Neumann Map. Let us
consider the following simple Dirichlet problem:(11)$\{\begin{array}{l}(-\triangle)^{\sigma}v=0 in\Omega v=0 on \mathbb{R}^{n}\backslash \Omega v(x)=v_{0}(x) non- negative and \dot{H}_{0}^{\sigma}- bounded\end{array}$with $0<\sigma<1$ . The fractional Laplacian of a function
expressed by the formula$f$:$\mathbb{R}^{n}arrow \mathbb{R}$is$(- \Delta)^{\sigma}f(x)=C_{n,\sigma}\int_{\mathbb{R}^{n}}\frac{f(x)-f(y)}{|x-y|^{n+2\sigma}}dy=C_{n,\sigma}\int_{\mathbb{R}^{n}}\frac{f(x+y)+f(x-y)-2f(x)}{|y|^{n+2\sigma}}dy$wheretion of$\bullet$$\bullet$$C_{n,\sigma}$

is some normalization constant. The Fourier transformaand we haveisand $(-\triangle)^{0}[u]=u$ .

$(-\triangle)^{\sigma}[u]$$|\xi|^{2\sigma}\hat{\text{\^{u}}}$$(-\Delta)^{\sigma_{1}}(-\triangle)^{\sigma_{2}}[u]=(-\Delta)^{\sigma_{1}+\sigma_{2}}[u]$If$\sigmaarrow 1$, then$(-\triangle)^{\sigma}[u]=-\Delta u$

In addition, the norm in$\dot{H}^{\sigma}$.

is given precisely by

.

(12)

$||f||_{H^{\sigma}}=\sqrt{\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2\sigma}}dxdy}$

That is equivalent to.

$||f||_{H^{\sigma}}\cong\sqrt{\int_{\mathbb{R}^{n}}|\xi|^{2\sigma}|f^{\wedge}(\xi)|^{2}d\xi}$for Fourier transform ofresults say that

$f$

in

$x$

$\dot{H}^{\sigma}\subset L^{2n/(n-2\sigma)}$space of. Note that the Sobolev embedding(Chap V in [St]). Indeed,$\dot{H}^{\sigma}$

is thefunctions for which (12) is integrable.
can be also thought as the normal derivative of some extension of (the Dirichlet to Neumann operator of ). See [CS] for ageneral discussion. We introduce first the corresponding extension
todefined fromby:$L^{2n/(n-2\sigma)}$$(-\triangle)^{\sigma}v$

$v$

$v$

$v^{*}$$C_{0}^{\infty}(\mathbb{R}^{n})$$C_{0}^{\infty}(\mathbb{R}^{n}\cross \mathbb{R}^{+})$$-\nabla(f\nabla v^{*})=0$$v^{*}(x,0)=v(x)$infor$R^{n}\cross(0,\infty)$$x\in R^{n}$for $a=1-2\sigma$ . Then the following result holds true: for defined on, we have:$v$$R^{n}$$(- \Delta)^{\sigma}v(x)=\partial_{v}v^{*}(x,0)=-\lim_{yarrow 0}fv_{y}^{*}(x, y)$73
NONL $\propto AL$ EQUATIONS

where we denoteboundary$\{y=0\}$$\partial_{v}v^{*}$.

the outward normal derivative of$v$‘

on the1.3. Outline. In this paper, we are going to summarize the recent developement on Nonlocal Nonlinear equations and its apphcation to
the geometric properties of the first eigen function of fractional Laplacian. In the section 2, we show the H\"older regularity and asymptotic
behavior of the dolutions for singular nonlocal equations. And in
the section 3, we show the the geometric properties of the first eigen
function of fractional Laplacian. Finally, at section 4, we summarize
the regularity theory on nonlocal fully nonlinear equations.
2. SINGULAR $N_{oN}$-LOCAL EQUATIONSIn this section, we consider initial value problem with fractional
fast diffusion, $[KsL1]$ :(13)$\{\begin{array}{l}(-\triangle)^{\sigma}u^{m}+u_{t}=0 in\Omega u=0 on \mathbb{R}^{n}\backslash \Omega u(x,0)=u_{0}(x) non- negative and \dot{H}_{0}^{\sigma}- bounded\end{array}$in the range of exponents, with $0<\sigma<1$ . In this work,
we will deal with a H\"older regularity of $v=u^{m}$ , which is a solutionof$\frac{n-2\sigma}{n+2\sigma}<m<1$(M.P)$\{\begin{array}{ll}(-\triangle)^{\sigma}v+(v^{\frac{1}{m}})_{t}=0 in\Omega v=0 on \mathbb{R}^{n}\backslash \Omega v(x,0)=v_{0}(x)=u_{0}^{m}(x) in \Omega,\end{array}$assuming that the imitial value is strictly positive in the interior ofin . Main two Theorems state as follows:$v_{0}$$\Omega$$\mathbb{R}^{n}$

Theorem 2.1. (Fromto
Let $v(x,t)$ be afunction in$L^{\frac{n-2\sigma}{n+20}}$$L^{\infty}$)$L^{\infty}(O,T,L^{\frac{2n}{n-2\sigma}(\Omega))}\cap L^{2}(0,T,\dot{H}_{0}^{\sigma}(\mathbb{R}^{n}))$, then$\sup_{x\in\Omega}|v(x, T)|\leq C^{*}\frac{||v_{0}||_{L^{2n}\overline{n}-T\sigma(\Omega)}}{T^{\frac{mn}{2mn-(n-20)(1+m)}}}$

for some constant $C’>0$ .
For the second theorem, we need better control ofTheorem 2.2 (H\"older regularity of fractional FDE).
, we defineFor
for $t_{0}>r^{2\sigma}>0$ . Assume now that, then there exist constants
bounded in$x_{0}=(x_{0}^{1}, \cdots,x_{0}^{n})$

$v$

.
,

$Q_{r}(x_{0},t_{0})=[x_{0}^{i}-r,x_{0}^{i}+r]^{n}\cross[t_{0}-r^{2\sigma},t_{0}]$and $v(x,t)$ is
and in $(0,1)$$[x_{0}^{i}-r,x_{0}^{i}+r]^{n}\subset\Omega$$\mathbb{R}^{n}\cross[t_{0}-r^{2\sigma},t_{0}]$$\gamma$$\beta$74
SUNGHOON KIM, YONG-CHEOL KIM, AND KI-AfM LEE

that can be detemined a priori only in terms of the data, such that is
.
in
$v$

$C^{\beta}$$Q_{\gamma r}(x_{0\prime}t_{0})$The details of the proofs can be found at $[KsL1]$ . In order to develop
the H\"older regularity method, it is necessary to localize the energy inequality by space and time truncation. Due to the non-locality of the
diffusion, this appears complicated. On the other hand,
can bethought as the normal derivative of some extension of (the Dirichlet
to Neumann operator of . See [CS] for a general discussion). This
allows us to realize the truncation as a standard local equation in one
more dimension: we introduce first the corresponding extension
by:todefined from$(-\Delta)^{\sigma}v$

$v$

$v$

$v$

$C_{0}^{\infty}(\mathbb{R}^{n})$$C_{0}^{\infty}(\mathbb{R}^{n}\cross \mathbb{R}^{+})$

infor$-\nabla(f\nabla v^{*})=0$$\mathbb{R}^{n}\cross(0,\infty)$$v^{*}(x,0)=v(x)$$x\in \mathbb{R}^{n}$for $a=1-2\sigma$ . (This extension consists simply in convolving with
the Poisson kemel of the upper half space in one more variable.)
Then the following result holds true: for defined on , we have:$v$$\mathbb{R}^{n}$$v$$(-\Delta)^{\sigma}v(\chi)=\partial_{v}v^{*}(x, 0)=-1\dot{m}fv_{y}^{*}(x, y)yarrow 0$

the outward normal derivative of on thewhere we denoteboundary $\{y=0\}$ . Hence, it is possible to consider the solution ofproblem (M.P) as the boundary value of which is solution of$v^{*}$$\partial_{v}v^{*}$$v$$v^{*}$(14)$\{\begin{array}{l}\nabla(f\nabla v^{*})=0 in y>0\lim_{yarrow 0}fv_{y}^{*}(x,y,t)=(v^{\frac{1}{m}})_{t}(x,0,t) \chi\in\Omega v^{*}(x,0,t)=0 on \mathbb{R}^{n}\backslash \Omega.\end{array}$Thus, we can obtain the H\"older estimate of immediately by showing
the H\"older regularity of .
$v$$v^{*}$

goes to infinity as
Since the diffusion coefficients$varrow 0$ , we need to control the oscillation of
from below. Hence, we
derived from such that $w^{*}(x,y,t)=$consider the new function$M-v^{*}(x,y,t+t_{0})$ with $M=M(t_{0})= \sup_{t\geq t_{0}>0}v^{*}$ . By Theorem 2.1, weknow that the solution satisfies$D(v)=|v|^{1-\frac{1}{m}}$$v$$w^{*}$$v^{*}(\cdot, t)\leq M(t_{0})<\infty$$v^{*}$$(t\geq t_{0})$.

75
NONLOCAL EQUATIONS

From this, we get to a familiar situation:$(15\{\begin{array}{l}\nabla(f\nabla w^{*})=0 in y>0-\lim_{yarrow 0^{+}}f\nabla_{y}w^{*}(x,y)=[(M-w^{*})^{\frac{1}{m}}]_{t}(x,0) x\in\Omega w^{*}(x,0,t)=M on \mathbb{R}^{n}\backslash \Omega.\end{array}$

The research is divided into four setps: at first step, we study
several properties of the Fast Diffusion Equation (shortly, FDE) withfractional powers(16)$(-\triangle)^{\sigma}u^{m}+u_{t}=0$,$( \frac{n-2\sigma}{n+2\sigma}<m<1)$.

More precisely, we explain Scale Invariance, -Contmction and Extinction on Finite Time. In second step, we show the existence of weaksolution of the problem (M.P). Also, we investigate the boundednessof the solutions of problem (M.P) for positive times. Lastly in thissection, we compute local energy inequality ofwhich will
be a key step in establishin local H\"older estimates. The proof of the
H\"older regularity of problems is given in third step. In this step, we
consider the extension of that solves (M.P). This allows us totreat non-linear problems, involving fractional Laplacians, as a local
problems. In the last section, we study the existence of $non-l\dot{m}$ear
eigenvalue problem with fractional powers which is asymptotic profile of the parabolic flow (M.P) on extinction time.
$L^{1}$$(w^{*}-k)_{\pm}$$g$$v^{*}$$v$

2.1. Properties of Fast Diffusion Equations with Fractional Powers.
Since the operator
converges toas the quantity goes to
1, it is natural to expect that the solutions of the equation (16) has
a lot in common with those of the FDE (of course, not in complete
accord). Hence, before coming to main issue, we will discuss suchproperties of FDE in this section.
$(-\triangle)^{\sigma}$$(-\triangle)$$\sigma$

2.1.1. Scale lnvariance. Let us examine the application of scaling transformations to the fractional powers of the FDE in some detail. Let$u=u(x,t)$ be a solution of the fractional powers of the FDE,
(17)$(-\triangle)^{\sigma}u^{m}+u_{t}=0$$(0<m<1)$ .

Then
(18)

$( \tau u)(x,t)=L^{\frac{2\sigma}{m-1}\tau^{-\frac{1}{m-1}}}u(\frac{x}{L},$ $\frac{t}{T})$.

is again a solution of the fractional powers of the FDE in the sameclass.

76
SUNGHOON KIM,

$YONGrightarrow CfflOL$

KIM, AND KI-AHM LEE

2.1.2. -contraction. This is a very important estimate which has
played a key role in the fractional powers of the FDE theory. It willallow us to develop existence, umiqueness and stabihty theory in the$L^{1}$

space .
$L^{1}$

withis a bounded domain ofLemma 2.3 ( -contraction). Letsmooth boundary, and let and be two smooth solutions of thefractional
powers of the fast dfffusion equation $(FDE)$ ;$L^{1}$$\Omega$$u$$\mathbb{R}^{n}$$\tilde{u}$$\{\begin{array}{l}(-\Delta)^{\sigma}u^{m}+u_{t}=0 in Q_{T}=\Omega\cross(0,T)u=0 on \mathbb{R}^{n}\backslash \Omega\end{array}$with initial date(19)

$u_{0},\tilde{u}_{0}$respectively. We have for every $t>\tau\geq 0$$\int_{\Omega}[u(x,t)-\tilde{u}(x,t)]_{+}dx\leq\int_{\Omega}[u(x,\tau)-\tilde{u}(x,\tau)]_{+}dx$

As a consequence,(20)$||u(t)-\tilde{u}(t)||_{1}\leq||u_{0}-\tilde{u}_{0}||_{1}$.

2.1.3. Extinction in Finite Time. The main difference with porous medium
equation is the finite time convergence of the solutions to the zero
solution, which replaces the infinite time stabilization that holds for$m\geq 1$ . Thisphenomenon is called extinction in finite time and read as
follows.
Lemma 2.4. If $u(x,t)$ is the$C^{2,1}$

fractional powers:

solution of the fast dfffusion equation with$\{\begin{array}{l}(-\Delta)^{\sigma}u^{m}+u_{t}=0 in Q_{\infty}=\Omega\cross(O,\infty)u=0 on \mathbb{R}^{n}\backslash \Omega u(x,0)=u_{0}(x)\in C^{0}(\Omega)\end{array}$is a bounded domain of with smooth boundary, then there exists$T^{*}>0$ such that $u(\cdot,t)=0$, i.e.,
for all

where$\Omega$$\mathbb{R}^{n}$$t\geq T^{*}$.

$\lim||u(, t)||_{\infty}=0$$tarrow T$

for some$T^{*}>0$. The solution can be continued past the extinction timein a weak sense as$u\equiv 0$$\Gamma$.

2.1.4. No waiting time. The main di&rence between local and nonlocal Porous Medium Equations is that there is no waiting time in
the nonlocal equation since
is equation to nonlocal integral
which is positive when the value ofis zero.
$u_{t}(x_{0},t_{0})$$u(x_{0}, t_{0})$77
NONLOCAL EQUATIONS

2.2. Weak solutions. First, we study the problem (M.P) in the classof nonnegative weak solutions. For the remainder of this paper weassume that$\frac{n-2\sigma}{n+2\sigma}<m<1$holds.
Definition 2.1. A non-negative weak solution ofequation $(M.P)$ is a locally, such that
integrable function,, and the identity
and $v=0$ on$v\in L_{loc}^{1}(\mathbb{R}^{n}\cross[0,\infty))$$[0,\infty))$(21)$(-\triangle)^{\sigma}v\in L_{loc}^{1}(\mathbb{R}^{n}\cross$$\mathbb{R}^{n}\backslash \Omega$$\int_{0}^{\infty}\int_{\Omega}v^{\frac{1}{m}}\eta_{t}dxdt=f_{0}^{\infty}\int_{\Omega}v[-(-\Delta)^{\sigma}\eta]dxdt$

holds for any testfunction$\eta\in C_{c}^{2,1}(\Omega\cross[0,\infty))$with $\eta=0$ on$\mathbb{R}^{n}\backslash \Omega$.

We show the existence and comparison result for weak solutions.
the proof is similar to the proof of the Theorem (5.5) in [Va].
Lemma 2.5. There exists a non-negative weak solution of $(M.P)$ . Moreover,the comparison princ\’iple holds for these solutions:
are weak solutions
with initial data such that. in , then. for all $t>0$ .
$v,\hat{v}$$v_{0}\leq\hat{v}_{0}a.e$$v\leq\hat{v}a.e$$\Omega$

2.3. Local Energy Estimate of . The key ingredients for H\"older
regularity are the Sobolev and local energy inequalities for the extension $w^{*}(x,y,t)=M-v^{*}(x,y,t+t_{0})$ with. The effectof the non-local part ofbecomes encoded locally in the extra
variable. The first result, Sobolev inequality, states as follows:$w^{*}$$M= \sup_{t\geq t_{0}>0}v^{*}$$(-\triangle)^{\sigma}$

Lemma 2.6 (Sobolev Inequality). For a cut-off function
supported in ,$\eta$

compactly$B_{r}$(22)$||\eta v||_{L^{\frac{2n}{n-20}}(\mathbb{R}^{n})}\leq C||\eta v||_{\dot{H}^{\sigma}(\mathbb{R}^{n})}$and$||\eta v||_{L^{2}(t_{1_{l}}t_{2_{l}}\cdot L^{2}(\mathbb{R}^{n}))}^{2}\leq$(23)$C(\sup_{t_{1}\leq t\leq t_{2}}||\eta v||_{L^{2}(\mathbb{R}^{n})}^{2}+||\nabla(\eta v)^{*}||_{L^{2}(t_{1\prime}t_{2\prime}\cdot L^{2}(B_{r\prime}^{*\gamma))}}^{2})|\{\eta v>0\}|^{\frac{2\sigma}{n+2\sigma}}$for some $C>0$ .
Lemma 2.7 (Local Energy Estimate).
Letbe such thatand letto (14) and let $w^{*}(x,y, t)=M-v^{*}(x,y,t+t_{0})$ withthere exists a constant such that for every$t_{1},$$t_{2}$$t_{1}<t_{2}$$\lambda$$v^{*}\in L^{\infty}(t_{1},t_{2};L^{2}(\mathbb{R}^{n}\cross \mathbb{R}^{+}))$

be solution$M= \sup_{t\geq t_{0}>0}v^{*}$$t_{1}\leq t\leq t_{2}$. Then,and cut-offfunction

78
SUNGHOON KIM, YONG-(HEOL KIM, AND KI-AHM LEE$\eta$

such that the restriction of $\eta(w^{*}-k)_{\pm}on$:$B_{r}^{*}is$compactly supported in$B_{r}\cross(-r,r)$$\frac{1}{m}\int_{B_{r}\cross\{t_{2}\}}\eta^{2}[\int_{0}^{w-k)_{\pm}}(M-k\mp\xi)^{\frac{1}{m}-1}\xi d\xi]dx$$+ \int_{t_{1}}^{t_{2}}\int_{B_{\dot{r}}}|\nabla(\eta(w-k)_{\pm})|^{2}$(24)

fdx dy dt$\leq f_{t_{1}}^{i_{2}}f_{B_{\dot{r}}}|(\nabla\eta)(w^{*}-k)_{\pm}|^{2}y^{a}dxdydt$$+ \frac{2}{m}\int_{t_{1}}^{t_{2}}\int_{B_{r}}[\int_{0}^{(w-k)_{*}}(M-k\mp\xi)^{\frac{1}{m}-1}\xi d\xi]|\eta\eta_{t}|dxdt$$+ \frac{1}{m}\int_{B_{r}\cross\{t_{1}\}}\eta^{2}[\int_{0}^{w-k)_{f}}(M-k\mp\xi)^{\frac{1}{m}-1}\xi d\xi]dx$

2.4. Key Lemmas.
Lemma 2.8. There exists positive numbers andsuch that if$\lambda$$\rho$

independent of$\mu^{\pm}and$$\omega$(25)$| \{(x,t)\in Q_{R}(\theta_{0}),w(x,t)>\mu^{+}-\frac{\omega}{2}\}|<\rho|Q_{R}(\theta_{0})|$then$w(x,t)< \mu^{+}-\frac{\lambda\omega}{4}$for all$(x,t)\in Q_{\frac{R}{2}}(\Theta_{0})$.

Idea of the proof Set, for any non-negative integer ,$k$$R_{k}= \frac{R}{2}+\frac{R}{2^{k+1}}$

We denote by$\tilde{B}_{R,\delta}^{*}$

the set,and$l_{k}=p^{+}- \lambda(\frac{\omega}{4}+\frac{\omega}{2^{k+2}})$$B_{R}\cross(0,\delta)$.

and introduce the cylinders$Q_{k}(\theta_{0})=B_{\overline{R}_{k}}\cross(-\theta_{0}^{-\alpha}R_{k}^{2\sigma},$

$0)$

and
$\tilde{Q}_{R_{k},\frac{\delta^{k}}{4}}^{*}(\theta_{0})=\tilde{B}_{R_{k},\frac{\delta^{k}}{4}}^{*}\cross(-\theta_{0}^{-\alpha}R_{k}^{2\sigma},0)$

Let
the

$\overline{w}(x,\tau)=w(x, \theta_{0}^{-\alpha}\tau)$$quantity\overline{Z}_{k}$and.

$\vec{w}(x,y, \tau)=w^{*}(x,y, \Theta_{0}^{-a}\tau)$

to be$\overline{Z}_{k}=|\{(x, t)\in Q_{R_{k}}(1):\overline{w}>l_{k}\}|$.

. We also define

79
NONLOCAL EQUATIONS

Then,will be equal toshow that there are$\overline{Z}_{k}$$Z_{k}$. Through the local energy estimate, wesuch that we have

$0<\sigma,$ $\delta<1$$\overline{Z}_{k+1}\leq C4^{2k}(\frac{4^{3-a}}{(1+a)R^{2}}+\frac{4^{5}\theta_{0}^{\alpha}}{2mM^{\alpha}R^{2\sigma}})\overline{Z}_{k-1}^{1+\frac{[be]}{n+2\sigma}}=C’4^{2k}\overline{Z}_{k-1}^{1+\frac{2\sigma}{n+2\sigma}}$If$\overline{Z}_{1}$is smalL thenconverges to zero, which implies the conclusion.

$\overline{Z}_{k}$$\square$Lemma 2.9 (Oscillation Lemma). There exist constants$\kappa\in(0,1)$ suchthat if$osc_{Q_{2R}^{*}}w^{*}=\omega=\mu^{+}-\mu^{-}$$\lambda^{*}>0$

and

,

then
$OSC_{Q_{R\tau}^{*}(\Theta_{0\prime}0,1-\frac{\rho}{2})^{w^{*}\leq\omega-\lambda^{*}=\kappa\omega}}$.

2.5. Asymptotic behaviour for the FDE with fractional powers.
The asymptotic description is based on the existence of appropriate solutions that serve as model for the behavior ear extinction:
there is asolution of the form
$n\backslash$$self-s\dot{m}\dot{u}lar$(26)$U(x,t,T)=(T-t)^{1/(1-m)}f(x)$for a certain profile $f>0$, where
linear elliptic equation$\varphi=f^{m}$$(- \triangle)^{\sigma}\varphi(x)=\frac{1}{1-m}\varphi(x)^{p}$is the solution of the super-,$p= \frac{1}{m}$such thatin with zero on. Hence, similarity meansin this case the separate-variables form. The existence and regularityof this solution depends on the exponent , indeed it exists for $p<$$(n+2\sigma)/(n-2\sigma)$ , the Sobolev exponent. Since $p=1/m$ ,this meansthat smooth separate-variables solutions exist for$\varphi>0$$\Omega$$\mathbb{R}^{n}\backslash \Omega$$p$$\frac{n-2\sigma}{n+2\sigma}<m<1$

an assumption that will be kept in the sequel. Note that the familyof solutions (26) has a free parameter $T>0$ .
The above family of solutions allows to describe the behavior ofgeneral solutions near their extinction time.
Theorem 2.10. Under the above assumptions on and , we have thefollowing property near the extinction time of a solution $u(x,t)$ : for any, we have a subsequence
sequenceasuch that$u_{0}$$\{u(x,t_{n})\}$$t_{n_{k}}arrow\tau*and$$m$$\varphi(x)$$\lim_{karrow\infty}(\Gamma-t_{n_{k}})^{-1/(1-m)}|u(x,t_{n_{k}})-U(x,t_{n_{k}}, \Gamma)|arrow 0$

80$SUNGH\infty N$ KIM,

YONG-CHEOL KM, AND KI-AHM LEE

uniformly in compact subset of for $U(x,t,T^{*})=(T^{*}-t)^{1/(1-m)}\varphi^{1/m}(x)$where is eigen-function offully nonlinear equation$\Omega$$\varphi$$a$$\{\begin{array}{ll}(-\Delta)^{\sigma}\varphi=\frac{1}{1-m}\varphi^{\frac{1}{m}} in \Omega\varphi=0 on \mathbb{R}^{n}\backslash \Omega\varphi>0 in \Omega.\end{array}$

3. GOEMETRIC PROPERTIES OF FlRST ElGENFUNCT10N
In this section we study convexity and concavity properties of nonlocal parabolic flows and derive related geometric properties for theasymptotic limmits of such evolutions. More precisely, we consider thenonnegative solution $u(x,t)$ of the Dirichlet problem for the fractional
power of Heat operator(27)$\{\begin{array}{l}u_{t}+(-\triangle)^{\frac{1}{2}}u=0 in\Omega u(x,t)=u_{0}(x)>0 in \Omega u> Oin \Omega u=0 on \mathbb{R}^{n}\backslash \Omega\end{array}$. In theposed on a strictly convex and bounded domain
renormalized hmit these flows converge to solutions of the nonlinear
eigen-value problems (28) below, especially $p=1$ . The method we
present produces power convexity results for these positive nonlinear
eigen-functions. As a second type of result, the evolution approachalso proves eventual power convexity in space for the solutions of
the parabolic problems. Eventual power convexity means that it willhold for large enough times even for data that are not initially power
convex.
$\Omega\in R^{n}$3.1. Elliptic Problems and History. Let us present the problems andsatisfy theconcepts to motivate our work. Let the function
following nonlinear eigenvalue problem:$\varphi(x)$(28)$\{\begin{array}{ll}(-\Delta)^{\frac{1}{2}}\varphi=\lambda\varphi^{p} in\Omega\varphi>0 in \Omega\varphi=0 on R^{n}\backslash \Omega.\end{array}$

The main question we address is motivated by the following conjecture:be the ground state
Conjecture 3.1 (cf. Conjecture 1.1 in [1]). Let
eigenfunction for the symmetric stable processes of index $0<\sigma<1$ killed
upon leaving the interval $I=(-1,1)$ . Then is concave on .
$\varphi_{1}^{\sigma}$$\varphi_{1}^{\sigma}$

$I$

81
NONLOCAL EQUATIONS

The functionis the first eigen function of (28) with being$(0<\sigma<1)$
replaced by. The only known case is when $\Omega=(-1,1)$, where the question is answered in the afflrmative in
and[BK]. More precisely, in [BK], Bafiuelos and Kulczycki consider the$n\geq 1$ , killed on
Cauchy process inleaving a bounded domain
S2, and investigate the eigenvalues
and eigenfunctionsof the. The method is quite different fromcorresponding generator in
the ones used for the Brownian motion which are not applicable inthis case due to the non-locality of the generator. The most detailed
information about the eigenvalues and eigenfunctions is found forthe case where $n=1$ and $\Omega=(-1,1)$ . In particular, it is proved thatis symmetric and concave on $(-1,1),$ is antisymmetric, concave
on $(0,1)$ and convex on $(-1,0)$ , each has at most $2n-2$ zeros on$\varphi_{1}^{\sigma}$$\frac{1}{2}$$\sigma,$$\sigma=\frac{1}{2}$$R^{n},$$\lambda_{n}$$\varphi_{n}$$L^{2}(\Omega)$$\varphi_{1}$$\varphi_{2}$$(-1,1)$$\varphi_{n}$.

In the particular case
it is easy to see that the operator
coincides with the Dirichlet to Neumann operator in the upper half
. More precisely, given $u(x)$ defined in , extend it tospace of$(-\triangle)^{\frac{1}{2}}$$\sigma=\frac{1}{2}$$\mathbb{R}^{n+1}$$u^{*}(x,y)$in$\mathbb{R}^{n}$$\mathbb{R}^{n+1}$

by convolving with the classical Poisson kemel.

3.2. Power-Convexity: Bounded Domain.
Lemma 3.2. There exist a weak solution
problem(29)

for$\psi\in\dot{H}^{\frac{1}{2}}(R^{n})$of the eigen-value$\{\begin{array}{ll}(-\triangle)^{\frac{1}{2}}\psi=\lambda\psi^{p} in\Omega\psi>0 in \Omega\psi=0 in R^{n}\backslash \Omega.\end{array}$$0<p< \frac{n+1}{n-1}$.

We address now the long-time geometrical properties of solutionsof the initial-value problem for the fractional power of the Heat Equation(30)$(-\triangle)^{\frac{1}{2}}u+u_{t}=0$

posed in a bounded domain
(31)

$u=0$

on

$\Omega$,with$\mathbb{R}^{n}\backslash \Omega$

,

$u>0$

in

$\Omega$and initial data(32)

$u(x,0)=u_{0}(x)\in\dot{H}^{\frac{1}{0^{2}}}(\Omega)$

Our geometrical results will be derived under the extra assumptionthat is strictly convex. It is easy to show that the fractional power ofthe Laplace operator has a countable discrete set of eigenvalues$\Omega$$\Sigma=$82

SUNGHOON KIM, YONG-CHEOL KIM, AND KI-AHM LEE$\{\lambda_{i}|\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}<\cdots\}$, whose eigen-functions$\{\phi_{n}\}$span,. Then,$\dot{H}^{\frac{1}{0^{2}}}(\Omega)$

is a normalized eigen-function corresponding tois the solution of the fractional powers of the Heat,
. On the other hand, for
operator with imitial data
. Hence,such that
there are coefficientswhere$\lambda_{n}$$\phi_{n}$$u_{n}(x,t)=e^{\lambda_{n}}{}^{t}\phi_{n}(x)$$u_{0}(x)\in\dot{H}^{\frac{1}{0^{2}}}(\Omega)$$\phi_{n}(x)$$u_{0}(x)= \sum_{n=1}^{\infty}a_{n}\phi_{n}(x)$$\{a_{n}\}$$u(x,t)= \sum_{n=1}^{\infty}a_{n}e^{-\lambda_{n}}{}^{t}\phi_{n}=a_{1}e^{-\lambda_{1}}{}^{t}\varphi+e^{-\lambda_{2}}{}^{t}\eta(x,t)$

where$||\eta(x,t)||_{L_{x}^{2}(R^{n})}<C<\infty$(33). Then$\varphi(x)$will be a solution of$\{\begin{array}{ll}(-\triangle)^{\frac{1}{2}}\varphi(x)=\lambda_{1}\varphi(x) in \Omega\varphi(x)=0 on \mathbb{R}^{n}\backslash \Omega.\end{array}$

We have the following results in the bounded domain, $[KsL2]$ .
Lemma 3.3 (Approximation Lemma). For every(34)$u_{0}\in\dot{H}^{\frac{1}{0^{2}}}(\Omega)$, we have$|e^{\lambda_{1}}{}^{t}u(x,t)-a_{1}\varphi(x)|\leq Ce^{-(\lambda_{2}-\lambda_{1})t}$

and
(35)

$||e^{\lambda_{1}}{}^{t}u(x,t)-a_{1}\varphi(x)||_{C_{\chi}^{k}(\Omega)}\leq CKe^{-(\lambda_{2}-\lambda_{1})t}$

for $k=1,2,$$\cdots$.

Corollary 3.4. If is convex, then the solution $u(x,t)$ of (30)-(32) is.
power-convex, i. e.,
Corollary 3.5. If is convex, then the stationary profile
of $u(x,t)$ is$\Omega$$D_{x}^{2}(u(x,t))^{-\frac{2}{n+1}}\geq 0$$\varphi(x)$$\Omega$

power-convex, i.e.,$D_{x}^{2}(\varphi(x))^{-\frac{2}{n+1}}\geq 0$.

3.3. Power-Convexity: Unbounded Domain. We now examin thesame geometrical questions for the Gauchy problem for the Fractional
Heat Equation
$e$

(36)

$\{\begin{array}{l}u_{t}+(-\Delta)^{\frac{1}{2}}u=0 in R^{n}\cross(0,\infty)u(x,0)=u_{0}(x) on R^{n}\end{array}$nonnegative, bounded, integrable and compactly
supported. Then, the solution of (36) is given bywith imitial data

$u_{0}$$u$$u(x,t)= \int_{\mathbb{R}^{n}}p(x,\xi,t)u_{0}(\xi)d\xi$

where$p(x, \xi,t)=\frac{c_{n}t}{(|x-\xi|^{2}+t^{2})^{\frac{n+1}{2}}}$,$t>0,$$x,\xi\in R^{n},$$c_{n}= \frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}$.

83

NONLOCAL EQUATIONSThe extension argument is also applied to Cauchy problem. Let
be the solution of the following extension problem:$u^{*}(x,z, t)$(37)$\{\begin{array}{l}\triangle_{x}u^{*}+u_{zz}^{*}=0 in R^{n}\cross R^{+}\cross(0,\infty)u_{z}^{*}(x,0,t)-u_{t}^{*}(x,0,t)=0 x\in \mathbb{R}^{n}u^{*}(x,0,0)=u_{0}(x) x\in R^{n}.\end{array}$

The details of proof of the following lemmas can be found at $[KsL2]$ .
Lemma 3.6. Let
be a continuous and bounded initialfunction witha compact support. If
is strictly convex, then the solution
ispower-convex in the space variable for all $t>0$, i.e.,.
The fact that solution of (36) is the trace of gives the following
Corollary.
Lemma 3.7. Let
be a continuous and bounded initialfunction witha compact support. If
is strictly convex, then the solution $u(x,t)$ of(36) is power-convex, i.e.,.
$u_{0}\geq 0$$(u_{0})^{-\frac{2}{n+1}}$$u^{*}$$x$$D_{\chi}^{2}[(u^{*})^{-\frac{2}{n+1}}]\geq 0$$u$$u^{*}$$u_{0}\geq 0$$(u_{0})^{-\frac{2}{n+1}}$$D_{x}^{2}(u(x,t))^{-\frac{2}{n+1}}\geq 0$

4. REGULARITY THEORY OF NONLOCAL FULLY NONLlNEAR EQUATIONSIn this section, we are going to discuss the recent development in
the regularity theory of nonlocal fully nonlinear equations.

4.1. Nonlocal Equations with nonsymmetric kemels. The conceptof viscosity solutions, its comparison principle and stability properties can be found in [CS] for symmetric kemels and in [KLl] forpossibly nonsymmetric kemels. Kim and Lee [KLl] considered muchlarger class of operators but prove the regularity of viscosity solutions
only for $1<\sigma<2$ .
Now we are going to consider a subclassofwhere the drift
effect created by the nonsymmetric kemel is controllable. Forand, we set$S^{\mathfrak{L}}$$S_{\eta}^{\mathfrak{L}}$$x\in B_{R}$$\varphi\in C_{B_{R}}^{2}(u,x)^{\pm}$.

$p_{R}(u,x,y,\cdot\nabla\varphi)=u(x+y)-u(x)-(\nabla\varphi(x)\cdot y)\chi_{B_{R}}(y)$For $u\in C^{1,1}[x]$ , we write $\mu_{R}(u,x,y)=\mu_{R}(u,x,y,\nabla u)$ . Then we define
andby replacing byin the definition. We note if $u\in C^{1,1}[x]$ , then.
p.v. $A_{n}\mu_{R}(u,x,y)K(y)dy$ . Key observations are the following:
For the nonsymmetric case, $K(y)$ and $K(-y)$ can be chosen any ofor. Therefore there could be an extra term$\mu_{R}^{\pm}$$\mathcal{M}_{\mathfrak{L}_{0},R}^{\pm}u(x,\nabla\varphi)$$\mathcal{M}_{\mathfrak{L}_{0}}^{\pm}u(x,\nabla\varphi)$$\mu_{R}$$\mu$$\mathcal{M}_{\mathfrak{L}_{0},R}^{\pm}u(x,\nabla\varphi)=\mathcal{M}_{\mathfrak{L}_{0},R}^{\pm}u(x)=$$\bullet$$\lambda/|y|^{n+\sigma}$$\Lambda/|y|^{n+\sigma}$$\int_{\mathbb{R}^{n}}\frac{|(\nabla u(x)\cdot y)\chi_{B_{1}}(y)|}{|y|^{n+\sigma}}dy$.

84$SUNGH\infty N$ KIM,

YONG-CHEOL KIM, AND KI-AHM LEE.
The equation is not scalmg invariant due toSomehow the equation has a drift term, not only the diffision
term. The case $1<\sigma<2$ and the case $0<\sigma\leq 1$ require different
technique due to the difference of the blow rate as approaches tozero and the decay rate as approaches to ifflinity. When $1<\sigma<2$,
a controllable decay rate of kemel allows H\"older regularities in alarger class, which is invariant under an one-sided scalin i.e. if isa solution of the homogeneous equation, then so isand supercritical case $(0<\sigma<1)$
for $0<e\leq 1$ . Critical casehave been studied in [BBC] with different techniques due to the slow.
decay rate of the kemel as$|\chi_{B_{1}}(y)|$$\bullet$$\bullet$

$|y|$

$|y|$

$u$

$g$

$u_{\epsilon}(x)=e^{-\sigma}u(ex)$$(\sigma=1)$$|x|arrow\infty$, where is a class of linear
Definition 4.1. Let $0<\eta\leq 1$ andif, for $R\in(O,1]$ ,integro-dffferential opemtors. Then we say that$R^{n}arrow R$ such that
there are(I)
is homogeneous of degree one, i.e. $B_{R}^{\pm}(0)=0$ and$t\in S^{\mathfrak{L}}$$\mathfrak{L}$$t\in S_{\eta}^{\mathfrak{L}}$$B_{R}^{\pm}:$$B_{R}^{\pm}(\nabla u)=$$B_{R}^{\pm}(\cdot)$$B_{R}^{+}( \frac{\nabla u}{|\nabla u|})|\nabla u|for|\nabla u|\neq 0$,(2) $|\{v\in S^{n-1} : B_{R}^{\pm}(v)<0\}|\geq\eta|S^{n-1}|>0$,(3)$\mathcal{M}_{\mathfrak{L},R\eta}^{-}u(x)\leq tu(x)-t0(x)\leq \mathcal{M}_{\mathfrak{L},R,\eta}^{+}u(x)$’$B(\mathbb{R}^{n})$

where

for

whenever $u\in C^{1,1}[x]\cap$$x\in B_{R\prime}$$\mathcal{M}_{\mathfrak{L},R,\eta}^{\pm}u(x):=\mathcal{M}_{\mathfrak{L},R}^{\pm}u(x)\pm B_{R}^{\pm}(\nabla u(x))\pm(2-\sigma)CR^{1-\sigma}|\nabla u(x)|$.

be a linear integro-differmtial opemtor with aDefinition 4.2. Let,
kernel K. For $0<R<1$ the drift vectorof at is $d\phi ned$ by$\mathcal{L}\in \mathfrak{L}$$b_{\mathcal{L},R}$$R$$\mathcal{L}$$b_{l,R}=(2-\sigma)\int_{B_{1}\backslash B_{R}}yK(y)dy$.

The details of proof of the following lemmas and theorems can befound at [KL2].
Lemma 4.1. Let $0<\sigma<2$ and $0<R<1$ . Let be a class of linear
integro-dffferential operators. Then we have the following results:(I) If $l$ is a linear integro-dffferential opemtor which is in , then$\mathfrak{L}$$\mathfrak{L}$$\mathcal{L}\in S_{\eta}^{\mathfrak{L}}$for$0< \eta\leq\frac{1}{2}$.

be a linear integro-dffferential operator with a kemel for$i=1,\cdots$ , N. Let
be the drift vectors of . Assume that there,
is a unit vector such that for any nonzero drift vector(2) Let

$K_{i}$$\mathcal{L}_{i}$$b_{L,R}$$\mathcal{L}_{i}$$a$$b_{\angle_{i},R}$$\{a,$ $\frac{b_{\mathcal{L}_{i\prime}R}}{|b_{L,R}|}\}>0$.

85

NONLOCAL EQUATIONS

If$I\in S^{\mathfrak{L}}$satisfies that$\min_{i=1,\cdots,N}\mathcal{L}_{i}u(x)\leq \mathcal{I}u(x)-\mathcal{I}0(x)\leq i,\cdots,N\max_{=1}\mathcal{L}_{i}u(x)$whenever$\eta>0$$u\in C^{1,1}[x]\cap B(\mathbb{R}^{n})$.

for$\chi\in B_{R}$, then$t\in S_{\eta}^{\mathfrak{L}}$for some

be a linear integro-dffferential operator with a kernel ,. Letbe the drift vectors of . Assume that there is avectorand $\eta>0$ such that for any nonzero drift vectors(3) Let$\mathcal{L}_{\alpha}$$K_{\alpha}$$\alpha\in I$$b_{L,R}$$\mathcal{L}_{\alpha}$$\mathfrak{a}\in S^{n-1}$$b_{\angle_{a\prime}R\prime}$$\{\mathfrak{a},$If$t\in S^{\mathfrak{L}}$for any$\frac{b_{L,R}}{|b_{X_{\alpha},R}|}\}\geq\eta^{\frac{1}{n-1}}$$\alpha\in I$.

satisfies that$\min_{\alpha\in I}\mathcal{L}_{\alpha}u(x)\leq tu(x)-$IO$(x) \leq\max \mathcal{L}_{\alpha}u(x)$$\alpha\in I$

whenever $u\in C^{1,1}[x]\cap B(R^{n})$ for$\chi\in B_{R}$, then we have$\mathcal{I}\in S_{\eta}^{\mathfrak{L}}$.

Lemma 4.2.
(I) Let $0<\sigma<2$ and let
be a class of linear integro-differential
with a symmetric kemel , thenopemtors. If
forsome $\eta\in(0,1]$ . In addition, if has symmetric kernel for all ,$\mathfrak{L}$$K$$l\in S^{\mathfrak{L}}$$\mathcal{L}\in S_{\eta}^{\mathfrak{L}}$$\mathcal{L}_{a}$

then

(2)

$\sup_{\alpha}\mathcal{L}_{a},\inf_{\alpha}.\mathcal{L}_{a}\in S_{\eta}^{\mathfrak{L}}$If $1<\sigma<2$, then$\alpha$forfor some$1\geq\eta>0$$S^{\mathfrak{L}_{0}}=S_{\eta}^{\mathfrak{L}_{0}}$.

$\eta\in(0,1]$.

At [KL2]. we have the following Hamack inequality and H\"older

regularity.

Theorem 4.3. Let
and assume thator$R\in(0,R_{0}]$$u\inB(R^{n})$that. If
is a positivefunction such that$\sigma\in(0,1]$$\sigma_{0}\in(1,2)$$\mathcal{M}_{\mathfrak{L}_{0}}^{-}u\leq\frac{C_{0}}{R^{\sigma}}$and$\mathcal{M}_{\mathfrak{L}_{0}}^{+}u\geq-\frac{C_{0}}{R^{\sigma}}$with$\sigma_{0}<\sigma<2$$\sigma\in(\sigma_{0},2]$

onan$B_{2R}$$or$$\mathcal{M}_{\mathfrak{L}_{0},R,\eta}^{-}u\leq\frac{C_{0}}{R^{\sigma}}$and$\mathcal{M}_{\mathfrak{L}_{0},R,\eta}^{+}u\geq-\frac{C_{0}}{R^{\sigma}}$with $0<\sigma\leq 1$ on$B_{2R}$in the viscosity sense, then there is uniform constant $C>0$ such that$\sup_{B_{R/2}}u\leq C(\inf_{B_{R/2}}u+C_{0})$Fordepends only on
depends only on$\sigma\in(\sigma_{0},2),$$\sigma\in(0,1],$

$C$

$C$

, the dimension , and$n,$, and .
$n$$\lambda,\Lambda$$\lambda,\Lambda,$.

$\sigma$$\eta$$\sigma_{0}$. And for86
SUNGHOON KIM, YONG$<HEOL$ KIM, AND KI-AHM LEE

Theorem 4.4. Let$\sigma_{0}\in(1,2)$

and assume thatthat $R\in(0,R_{0}]$ . If is a boundedfunction on$u$$\mathcal{M}_{\mathfrak{L}_{0}}^{-}u\leq\frac{C_{0}}{R^{\sigma}}$and$\mathcal{M}_{\mathfrak{L}_{0}}^{+}u\geq-\frac{C_{0}}{R^{\sigma}}$$\sigma\in(0,1]$$R^{n}$

withor$\sigma\in(\sigma_{0},2]$

andsuch that$\sigma_{0}<\sigma<2$on$B_{2R}$$or$$\mathcal{M}_{\mathfrak{L}_{0},R,\eta}^{-}u\leq\frac{C_{0}}{R^{\sigma}}$and$\mathcal{M}_{\mathfrak{L}_{0},R,\eta}^{+}u\geq-\frac{C_{0}}{R^{\sigma}}$with $0<\sigma\leq 1$ onin the viscosity smse, then there is some constant$\alpha>0$$B_{2R}$such that$||u||_{C^{a}(B_{R/2})} \leq\frac{C}{R^{\alpha}}(||u||_{L^{\infty}(\mathbb{R}^{n})}+C_{0})$and depend
where $C>0$ is some universal constant. For, the dimension , and . And for $\sigma\in(0,1],$ and depend
only on, and .
only on$\sigma\in(\sigma_{0},2),$$n$$\lambda,\Lambda$$\lambda,\Lambda,$$n,$$\sigma$$\sigma_{0}$$\alpha$$\alpha$$C$$C$$\eta$

4.2. Nonlocal Nonlinear Parabolic Equations with symmetric kernels. Sinuilar regularity theory have been discussed on parabolic
nonlocal nonlinear equation at [KL3]. Key observations are the following:
The equation is local in time while it is nonlocal in the spacevariable. Caffarelli and Silvestre considered a sequence of dyadic
rings in space at A-B-P estimate to find the balance of quantities in
the integral. But a simple generalization of the ring in space to one inspace-time fails since the equation is local in the time variable. Suchunbalance between local and nonlocal terms in the equation requires
estimate.
more fine analysis to find a parabolic version of
There is a time delay to control the lower bound in a smallneighborhood of a point by the current value at the point, which is amain difference between elliptic and parabolic equations. The detailsof results can be found at [KL3]Acknowledgement. Ki-Ahm Lee was supported by Basic Science
Research Program through the National Research Foundation of
Korea(NRF) grant funded by the Korea govemment(MEST)(2010$\bullet$$A-Brightarrow P$$\bullet$

0001985).

REFERENCES[BBC] Martin T. Barlow; RiChard F. Bass; Zhen-Qing Chen; Moritz Kassmam,

Non-local D irichlet forms and symmetric jump processes, Trams. Amer. Math. Soc.

361, 2009, no. 4, 1963-1999.
[BK] Richard F. Bass, Moritz Kassmam, Holder continuity of harmonicfunctions withrespect to operators of variable order, Comm. Partial Di&rential Equations 30(2005), 1249-1259

87
NONLOCAL EQUATIONS[1] Rodrigo Bar uelos, Tadeusz Kulczycki, Pedro J. M\’endez-Hern\’andez. On the
shape of the ground state eigenfunctionfor stable processes. Potential Anal. 24(2006),

no. 3[CS] Luis A. Caffarelli and Luis Silvestre, Regularity theoryforfully nonlinear integrodifferential equations, Comm. Pure Appl. Math. 62(5), 2009, 597-638.
[KLl] Yong-Cheol Kim and Ki-Ahm Lee, Regularity resultsforfully nonlinear integrooperators with nonsymmetric positive kemels: Subcritical Case, submit-

differential

ted[KL2] Yong-Cheol Kim and Ki-Ahm Lee, Regularity resultsforfully nonlinear integrodifferential operators with nonsymmetric positive kernels, submitted[KL3] Yong-Cheol Kim and Ki-Ahm Lee, Regularity results forfully nonlinear parabolic integro-dffferential operators $http://arxiv.oryabs/1109.3807$[KsLl] Sunghoon Kim and Ki-Ahm Lee, Hlder Estimates for Singular Non-local Parabolic Equations, Journal of Functional Analysis, 261, (2011) 3482-3518.
[KsL2] Sunghoon Kim and Ki-Ahm Lee, Ground State Eigenfunction
for stable$\alpha-$processes $http://arxiv.oryabs/1105.3283$

Luis Silvestre, Holder estimates for solutions of integro-differential equations likethefractional laplace, Indiana Univ. Math. Jour. 55(3), 2006, 1155-1174.
[St] E. M. Stein, Singular integrals and differentiability properties
offunctions. Princeton
Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
[Va] V\’azquez, J. L.: The Porous Medium Equation. Mathematical Theory, Oxford
Mathematical Monographs. The Clarendon Press, Oxford University Press,
Oxford, 2007.
[S]

INSTITUTE OF MATHEMATICS, ACADBMIA SINICA, TAIPEI, 10617, TAIWAN, R.O.C.
E-mail address: gauss79@math. sinica. edu. tw

DEPARTMENT OF MATHEMATICS EDUCATION, KOREA UNIVERSITY, SEOUL 136-701,
KOREA
E-mail address: ychkim@korea. ac. kr

SEOUL NATIONAL UNIVERSITY, SEOUL 151-747, KOREA
E-mail address: kiahm@snu.ac.kr