Monster Pucci operator and Fractional Laplacian: Difference between pages
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The fractional Laplacian $(-\Delta)^s$ is a classical operator which can be defined in several equivalent ways. | |||
It is the most typical elliptic operator of order $2s$. | |||
== Definitions == | |||
=== 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. | |||
Revision as of 01:25, 8 June 2011
The fractional Laplacian $(-\Delta)^s$ is a classical operator which can be defined in several equivalent ways.
It is the most typical elliptic operator of order $2s$.
Definitions
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.