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imported>Luis |
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| 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 $u \in \mathcal{S}(\mathbb{R}^n)$ we can write the operator as | | The Monge-Ampere equation refers to |
| | \[ \det D^2 u = f(x). \] |
| | The right hand side $f$ is always nonnegative. The unknown function $u$ should be convex. |
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| \[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\] | | The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator |
| | \[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\] |
| | is concave. |
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| 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.
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| == Definitions ==
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| All the definitions below are equivalent.
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| === As a pseudo-differential operator ===
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| The fractional Laplacian is the pseudo-differential operator with symbol $|\xi|^{2s}$. In other words, the following formula holds
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| \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\]
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| for any function (or tempered distribution) for which the right hand side makes sense.
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| 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.
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| === From functional calculus ===
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| 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 $(-\Delta)^s$.
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| This definition is not as useful for practical applications, since it does not provide any explicit formula.
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| === As a singular integral ===
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| If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula
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| \[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\]
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| Where $c_{n,s}$ is a constant depending on dimension and $s$.
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| This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.
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| === As a generator of a [[Levy process]] ===
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| 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
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| \[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]
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| This definition is important for applications to probability.
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| == Inverse operator ==
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| 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'':
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| \[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\]
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| which holds as long as $u$ is integrable enough for the right hand side to make sense.
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| == Heat kernel ==
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| The fractional heat kernel $p(t,x)$ is the fundamental solution to the [[fractional heat equation]]. It is the function which solves the equation
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| \begin{align*}
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| p(0,x) &= \delta_0 \\
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| p_t(t,x) + (-\Delta)^s p &= 0
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| \end{align*}
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| 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 $C$
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| \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]
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| Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
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| \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]
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| == Poisson kernel ==
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| Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem
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| \begin{align*}
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| u(x) &= g(x) \qquad \text{if } x \notin B_1 \\
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| (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1.
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| \end{align*}
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| The solution can be computed explicitly using the Poisson kernel
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| \[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
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| where<ref name="R"/>
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| \[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]
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| The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')<ref name="L"/>.
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| == Green's function for the ball ==
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| For a function $g \in L^2(B_1)$, there exists a unique function $u \in H^s(\R^n)$ such that
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| \begin{align*}
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| u(x) &= 0 && \text{if } x \notin B_1 \\
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| (-\Delta)^s u &= g(x) && \text{if } x \in B_1.
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| \end{align*}
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| The solution is given explicitly using the Green's function,
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| \[ u(x) = \int_{B_1} G_{B_1}(x, y) g(y) \mathrm d y, \]
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| where<ref name="R"/>
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| \[ G_{B_1}(x, y) = C_{n,s} |x - y|^{2 s - n} \int_0^{r_0(x, y)} \frac{r^{s-1}}{(r+1)^{n/2}} \, \mathrm d r \]
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| with
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| \[ r_0(x, y) = \frac{(1 - |x|^2) (1 - |y|^2)}{|x - y|^2} . \]
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| The above formula holds for all $s \in (0, 1)$ also for $n = 1$.<ref name="BGR"/>
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| == Regularity issues ==
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| Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This follows from the smoothness of the Poisson kernel for balls.
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| === Full space regularization of the Riesz potential ===
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| 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 $u \in B_{p,q}^{r+2s}$, $s>0$. However, if $f$ belongs to $L^p$ then it does not follow that $u\in W^{2s,p}$; this is true only for $p\geq2$. For $1<p<2$ one only have $u\in B^{2s}_{p,2}\supset W^{2s,p}$ ---see Chapter V in Stein<ref name="Stein"/>.
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| === Boundary regularity ===
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| 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).
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| Even if $u$ is not $C^\infty$ up to the boundary, we have the following: consider the solution $u$ to the Dirichlet problem
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| \[\left\{ \begin{array}{rcll}
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| (-\Delta)^s u &=&g&\textrm{in }\Omega \\
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| u&=&0&\textrm{in }\R^n\backslash \Omega.
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| \end{array}\right.\]
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| If $\Omega$ is $C^\infty$, then
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| \[g\in C^\infty(\overline\Omega)\qquad \Longrightarrow \qquad u/d^s\in C^\infty(\overline\Omega),\]
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| where $d(x)$ is (a smoothed version of) the distance to $\partial\Omega$; see <ref name="Grubb"/> and also <ref name="RS"/>. Moreover, when $g$ is $C^\alpha$ and $\Omega$ is $C^\infty$, then $u/d^s$ is $C^{\alpha+s-\epsilon}$, and when $g\in L^\infty$ then $u/d^s$ is $C^{s-\epsilon}$.
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| Related to this, if $g$ is not bounded but only in $L^p(\Omega)$ then $u\in L^q$ with $q=\frac{np}{n-2ps}$ in case $p<n/(2s)$, while $u\in L^\infty(\Omega)$ in case $p>n/(2s)$ ---see for example Proposition 1.4 in <ref name="RS2"/>.
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| == References ==
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| {{reflist|refs=
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| <ref name="BGR">{{Citation | last1=Blumenthal | first1=R. M. | last2=Getoor | first2=R. K. | last3=Ray | first3=D. B. | title=On the distribution of first hits for the symmetric stable processes | url=http://www.jstor.org/stable/1993561 | year=1961 | journal=Trans. Amer. Math. Soc. | issn=0002-9947 | volume=99 | pages=540–554}}</ref>
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| <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>
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| <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>
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| <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>
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| <ref name="RS">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The Dirichlet problem for the fractional Laplacian: regularity up to the boundary | url=http://arxiv.org/abs/1207.5985 | year=2012 | journal=[[J. Math. Pures Appl.]] | volume=101 pages=275-302 }}</ref>
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| <ref name="Stein">{{Citation | last1=Stein | first1=E. | title=Singular Integrals And Differentiability Properties Of Functions | publisher=[[Princeton Mathematical Series]] | year=1970}}</ref>
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| <ref name="RS2">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The extremal solution for the fractional Laplacian | url=http://arxiv.org/abs/1305.2489 | year=2013 | journal=[[Calc. Var. Partial Differential Equations]] | pages=to appear }}</ref>
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| <ref name="Grubb">{{Citation | last1=Grubb | first1=G. | title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators | url=http://arxiv.org/abs/1310.0951 | year=2014 | journal=[[arXiv]] | pages=1-43 }}</ref>
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