Aleksandrov-Bakelman-Pucci estimates and Fractional Laplacian: Difference between pages

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The celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated often as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations. It is the backbone of the regularity theory of fully nonlinear second order elliptic equations <ref name="CC"/> and more recently for [[Fully nonlinear integro-differential equations]] <ref name="CS"/>.  
The fractional Laplacian $(-\Delta)^s$ is a classical operator which gives the standard Laplacian when $s=1$. One can think of $-(-\Delta)^s$ as the most basic  [[elliptic linear integro-differential operator]] of order $2s$ and can be defined in several equivalent ways (listed below). A range of powers of particular interest is $s \in (0,1)$, in which case for $u \in \mathcal{S}(\mathbb{R}^d)$ we can write the operator as


== The classical Alexsandroff-Bakelman-Pucci Theorem ==  
\[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\]


Let $u$ be a viscosity supersolution of the linear equation:
where $c_{n,s}$ is a universal constant and $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$. This particular expression shows that in this range of $s$ the operator enjoys the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)^s u(x) \geq 0$, with equality only if $u$ is constant. From this monotonicity, a [[comparison principle]] can be derived for equations involving the fractional Laplacian.


\[ Lu \leq f(x) \;\; x \in B_1\]
== Definitions ==
\[ u \leq 0 \;\; x \in \partial B_1\]
\[ Lu:=a_{ij}(x) u_{ij}(x)\]


The coefficients $a_{ij}(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have
All the definitions below are equivalent.


\[ \lambda |\xi|^2 \leq a_{ij}(x) \xi_i\xi_j \leq \Lambda |\xi|^2 \;\;\forall \; \xi \in \mathbb{R}^n \]
=== As a pseudo-differential operator ===
The fractional Laplacian is the pseudo-differential operator with symbol $|\xi|^{2s}$. In other words, the following formula holds
\[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\]
for any function (or tempered distribution) for which the right hand side makes sense.


Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that
This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.


\[ \sup \limits_{B_1}\; u^n \leq C_{n,\lambda,\Lambda} \int_{\{ u=\Gamma_u \} } f_+^n dx \]
=== From functional calculus ===
Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.


Where $\Gamma_u$ is the "concave envelope" of $u$: it is the smallest non-nonnegative concave function in $B_2$ that lies above $u$ in $B_1$. The fact that the integration on the right hand side takes place only on the set where $u$ agrees with its concave envelope is an important feature of the estimate and it is not to be overlooked <ref name="CC"/>.
This definition is not as useful for practical applications, since it does not provide any explicit formula.


== ABP-type estimates for integro-differential equations ==
=== As a singular integral ===
If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula
\[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\]


The setting for integro-differential equations is similar, what changes are the operators: let $u$ be a viscosity supersolution of the equation
Where $c_{n,s}$ is a constant depending on dimension and $s$.


\[ Lu \leq f(x) \;\; x \in B_1\]
This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.
\[ u \leq 0 \;\; x \in  B_1^c\]
\[ Lu:= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{a(x,y)}{|y|^{n+\sigma}} dy\]


Here $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$ and $\sigma \in (0,2)$. The function $a(x,y)$ is only assumed to be measurable and such that for some $\Lambda\geq\lambda>0$ we have
=== As a generator of a [[Levy process]] ===
The operator can be defined as the generator of $\alpha$-stable Lévy processes. More precisely, if $X_t$ is the isotropic $\alpha$-stable Lévy process starting at zero and $f$ is a smooth function, then
\[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]


\[ \lambda \leq  a(x,y) \leq  \Lambda \;\;\forall\;x,y\in \R^n\]
This definition is important for applications to probability.


As in the second order ABP, the function $f$ is assumed to be continuous. Then, Caffarelli and Silvestre proved <ref name="CS"/> there is an estimate
== Inverse operator ==
The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the ''Riesz potential'':
\[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\]
which holds as long as $u$ is regular enough for the right hand side to make sense.


\[ \sup \limits_{B_1} \; u^n \leq  C_{n,\lambda,\Lambda,\sigma}\sum \limits_{i=1}^m  ( \sup \limits_{Q_i^*} |f|^n) |Q^*_i|  \]
== Heat kernel ==
The fractional heat kernel $p(t,x)$ is the fundamental solution to the [[fractional heat equation]]. It is the function which solves the equation
\begin{align*}
p(0,x) &= \delta_0 \\
p_t(t,x) + (-\Delta)^s p &= 0
\end{align*}


For some finite collection of non-overlapping cubes $\{Q_i \}_{i=1}^m$ that cover the set $\{ u=\Gamma_u\}$, each cube having non-zero intersection with this set.  Moreover, all the cubes have diameters $d_i \lesssim 2^{-\frac{1}{2-\sigma}}$. As before, $\Gamma_u$ denotes the "concave envelope" of $u$ in $B_2$.
The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables, but the following inequalities are known to hold for some constant $C$
\[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]


Furthermore, although the sharp constant may depend on $\sigma$, it is uniformly bounded for all $\sigma$ bounded away from zero. In particular, as $\sigma \to 2^-$ the above constant does not blow up, and since the diameter of the cubes goes to zero as $\sigma$ approaches $2$, one can check that this last estimate implies the second order ABP in the limit.
Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
\[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]


== References ==  
== Poisson kernel ==
Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem
\begin{align*}
u(x) &= g(x) \qquad \text{if } x \notin B_1 \\
(-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1.
\end{align*}


The solution can be computed explicitly using the Poisson kernel
\[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
where<ref name="R"/>
\[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]
The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')<ref name="L"/>.
== Regularity issues ==
Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This is a classical fact for pseudo-differential operators{{citation needed}}.
=== Full space regularization of the Riesz potential ===
If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions ''up to $2s$ derivatives''. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of <ref name="S"/>, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$.
=== Boundary regularity ===
From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel).
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="CC">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Cabré | first2=Xavier | title=Fully nonlinear elliptic equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-0437-7 | year=1995 | volume=43}}</ref>
<ref name="L">{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1972}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="R">{{Citation | last1=Riesz | first1=M. | title=Intégrales de Riemann-Liouville et potentiels | url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article | year=1938 | journal=Acta Sci. Math. Szeged | issn=0001-6969 | volume=9 | issue=1 | pages=1–42}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
}}
}}

Revision as of 05:06, 18 July 2012

The fractional Laplacian $(-\Delta)^s$ is a classical operator which gives the standard Laplacian when $s=1$. One can think of $-(-\Delta)^s$ as the most basic elliptic linear integro-differential operator of order $2s$ and can be defined in several equivalent ways (listed below). A range of powers of particular interest is $s \in (0,1)$, in which case for $u \in \mathcal{S}(\mathbb{R}^d)$ we can write the operator as

\[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\]

where $c_{n,s}$ is a universal constant and $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$. This particular expression shows that in this range of $s$ the operator enjoys the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)^s u(x) \geq 0$, with equality only if $u$ is constant. From this monotonicity, a comparison principle can be derived for equations involving the fractional Laplacian.

Definitions

All the definitions below are equivalent.

As a pseudo-differential operator

The fractional Laplacian is the pseudo-differential operator with symbol $|\xi|^{2s}$. In other words, the following formula holds \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\] for any function (or tempered distribution) for which the right hand side makes sense.

This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.

From functional calculus

Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.

This definition is not as useful for practical applications, since it does not provide any explicit formula.

As a singular integral

If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula \[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\]

Where $c_{n,s}$ is a constant depending on dimension and $s$.

This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.

As a generator of a Levy process

The operator can be defined as the generator of $\alpha$-stable Lévy processes. More precisely, if $X_t$ is the isotropic $\alpha$-stable Lévy process starting at zero and $f$ is a smooth function, then \[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]

This definition is important for applications to probability.

Inverse operator

The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the Riesz potential: \[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\] which holds as long as $u$ is regular enough for the right hand side to make sense.

Heat kernel

The fractional heat kernel $p(t,x)$ is the fundamental solution to the fractional heat equation. It is the function which solves the equation \begin{align*} p(0,x) &= \delta_0 \\ p_t(t,x) + (-\Delta)^s p &= 0 \end{align*}

The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables, but the following inequalities are known to hold for some constant $C$ \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]

Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]

Poisson kernel

Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem \begin{align*} u(x) &= g(x) \qquad \text{if } x \notin B_1 \\ (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1. \end{align*}

The solution can be computed explicitly using the Poisson kernel \[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\] where[1] \[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]

The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')[2].

Regularity issues

Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This is a classical fact for pseudo-differential operators[citation needed].

Full space regularization of the Riesz potential

If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions up to $2s$ derivatives. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of [3], but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$.

Boundary regularity

From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel).

References