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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 $u \in \mathcal{S}(\mathbb{R}^d)$ we can write the operator as
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| \[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\]
| | The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates<ref name="MR0097227"/>. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain, |
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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 regular 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*} | | \begin{align*} |
| p(0,x) &= \delta_0 \\
| | \Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\ |
| p_t(t,x) + (-\Delta)^s p &= 0
| | \frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega |
| \end{align*} | | \end{align*} |
| | The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side). |
| | Particular solutions are given for instance by the planar profiles |
| | \[ |
| | P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0 |
| | \] |
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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 $C$ | | The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one. |
| \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]
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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 == | | Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives |
| Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem
| | \[ |
| | \frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2 |
| | \] |
| | By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies |
| \begin{align*} | | \begin{align*} |
| u(x) &= g(x) \qquad \text{if } x \notin B_1 \\
| | \Delta w &= 0 \text{ in } \{x_n>0\}\\ |
| (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1.
| | \partial_t w &= a\partial_n w \text{ on } \{x_n=0\} |
| \end{align*} | | \end{align*} |
| | Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$, |
| | \[ |
| | \partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w |
| | \] |
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| The solution can be computed explicitly using the Poisson kernel
| | == References == |
| \[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
| | {{reflist|refs= |
| 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"/>. | | <ref name="MR0097227">{{Citation | last1=Saffman | first1= P. G. | last2=Taylor | first2= Geoffrey | title=The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid | journal=Proc. Roy. Soc. London. Ser. A | issn=0962-8444 | year=1958 | volume=245 | pages=312--329. (2 plates)}}</ref> |
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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 is a classical fact for pseudo-differential operators{{citation needed}}.
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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 $u \in B_{p,q}^{r+2s}$.
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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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| == References ==
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| {{reflist|refs=
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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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| }} | | }} |
This article is a stub. You can help this nonlocal wiki by expanding it.
The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates[1]. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain,
\begin{align*}
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega
\end{align*}
The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side).
Particular solutions are given for instance by the planar profiles
\[
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
\]
The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.
Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives
\[
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
\]
By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies
\begin{align*}
\Delta w &= 0 \text{ in } \{x_n>0\}\\
\partial_t w &= a\partial_n w \text{ on } \{x_n=0\}
\end{align*}
Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$,
\[
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w
\]
References
- ↑ Saffman, P. G.; Taylor, Geoffrey (1958), "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid", Proc. Roy. Soc. London. Ser. A 245: 312--329. (2 plates), ISSN 0962-8444