imported>Adina |
imported>Nestor |
| Line 1: |
Line 1: |
| == Well posedness of the supercritical [[surface quasi-geostrophic equation]] ==
| | The fractional Laplacian $-(-\Delta)^s$ $s>0$ 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 |
| Let $\theta_0 : \R^2 \to \R$ be a smooth function either with compact support or periodic. Let $s \in (0,1/2)$. Is there a global classical solution $\theta :\R^2 \to \R$ for the SQG equation?
| |
| \begin{align*}
| |
| \theta(x,0) &= \theta_0(x) \\
| |
| \theta_t + u \cdot \nabla \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
| |
| \end{align*}
| |
| where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
| |
| | |
| This is a very difficult open problem. It is believed that a solution would be a major step towards the understanding of Navier-Stokes equation. In the supercritical regime $s\in (0,1/2)$, the effect if the drift term is larger than the diffusion in small scales. Therefore, it seems unlikely that a proof of well posedness could be achieved with the methods currently known and listed in this wiki.
| |
| | |
| Note that if the relation between $u$ and $\theta$ was changed by $u = R\theta$, then the equation is ill posed. This suggests that the divergence free nature of $u$ must play an important role, unlike the critical and subcritical cases $s \geq 1/2$.
| |
| | |
| == Regularity of [[nonlocal minimal surfaces]] ==
| |
|
| |
|
| A nonlocal minimal surface that is sufficiently flat is known to be smooth <ref name="CRS"/>. The possibility of singularities in the general case reduces to the analysis of a possible existence of nonlocal minimal cones. The problem can be stated as follows.
| | \[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\] |
|
| |
|
| For any $s \in (0,1)$, and any natural number $n$, is there any set $A \in \R^n$, other than a half space, such that
| | 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 the operator enjoys a [[comparison principle]], since if $u\leq v$ everywhere with equality a some point $x$ then $\delta u(x,y) \leq \delta v(x,y)$ for all $y$, implying that |
| # $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
| |
| # If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds
| |
| \[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]
| |
|
| |
|
| When $s$ is sufficiently close to one, such set does not exist if $n < 8$.
| | \[ (-\Delta)^su(x) \geq (-\Delta)^sv(x) \] |
|
| |
|
| == An integral ABP estimate == | | == Definitions == |
|
| |
|
| The nonlocal version of the [[Alexadroff-Bakelman-Pucci estimate]] holds either for a right hand side in $L^\infty$ <ref name="CS"/> (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels <ref name="GS"/>. Would the following result be true?
| | All the definitions below are equivalent. |
|
| |
|
| Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
| | === As a pseudo-differential operator === |
| \[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
| | The fractional Laplacian is the pseudo-differential operator with symbol $|\xi|^{2s}$. In other words, the following formula holds |
| Where $\chi_{A_n}$ stands for the characteristic function of the sets $A_n$. Assume that the kernels $K$ satisfy symmetry and a uniform ellipticity condition
| | \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\] |
| \begin{align*} | | for any function (or tempered distribution) for which the right hand side makes sense. |
| K(x,y) &= K(x,-y) \\
| |
| \lambda |y|^{-n-s} \leq K(x,y) &\leq \Lambda |y|^{-n-s} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2).
| |
| \end{align*}
| |
| If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?
| |
|
| |
|
| This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.<ref name="GS"/> | | 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. |
| | |
| == A [[comparison principle]] for $x$-dependent nonlocal equations which are '''not''' in the Levy-Ito form ==
| |
| Consider two continuous functions $u$ and $v$ such that
| |
| \begin{align*}
| |
| u(x) &\leq v(x) \qquad \text{for all $x$ outside some set } \Omega,\\
| |
| F(x,\{I_\alpha u(x)\}) &\geq F(x,\{I_\alpha v(x)\})\qquad \text{for all $x \in \Omega$}.
| |
| \end{align*}
| |
| Is it true that $u \leq v$ in $\Omega$ as well?
| |
|
| |
|
| It is natural to expect this result to hold if $F$ is continuous respect to $x$ and the [[linear integro-differential operators]] $I_\alpha$ satisfy some nondegeneracy condition and continuity respect to $x$. Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.<ref name="BI"/>
| | === From functional calculus === |
|
| |
|
| == A local [[differentiability estimates|$C^{1,\alpha}$ estimate]] for integro-differential equations with nonsmooth kernels ==
| | 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 $(-\Delta)^s$. |
|
| |
|
| Assume that $u : \R^n \to \R$ is a bounded function satisfying a [[fully nonlinear integro-differential equation]] $Iu=0$ in $B_1$. Assume that $I$ is elliptic with respect to the family of kernels $K$ such that
| | This definition is not as useful for practical applications, since it does not provide any explicit formula. |
| \[ \frac{\lambda(2-s)}{|y|^{n+s}} \leq K(y) \leq \frac{\Lambda(2-s)}{|y|^{n+s}}. \]
| |
| Is it true that $u \in C^{1,\alpha}(B_1)$?
| |
|
| |
|
| An extra symmetry assumptions on the kernels may or maynot be necessary. The difficulty here is the lack of any smoothness assumption on the tails of the kernels $K$. This assumption is used in a localization argument in the proof of the [[differentiability estimates|$C^{1,\alpha}$ estimates]] <ref name="CS"/>. It is conceivable that the assumption may not be necessary at least for $s>1$.
| | === 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 need of the smoothness assumption for the $C^{1,\alpha}$ estimate is a subtle technical requirement. It is easy to overlook going through the proof naively.
| | Where $c_{n,s}$ is a constant depending on dimension and $s$. |
|
| |
|
| Note that the assumption is used only to localize an iteration of the [[Holder estimates]]. An equation of the form $Iu = f$ in the whole space $\R^n$ with $f$ smooth enough would easily have $C^{1,\alpha}$ estimates without any smoothness restriction of the tails of the kernel.
| | This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems. |
|
| |
|
| It is not clear how important or difficult this problem is. The solution may end up being a relatively simple technical approximation technique or may require a fundamentally new idea.
| | === As a generator of a [[Levy process]] === |
| | The operator can be defined as the generator of $\alpha$-stable Levy processes. More precisely, if $X_t$ is an $\alpha$-stable process starting at zero and $f$ is a smooth function, then |
| | \[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \] |
|
| |
|
| The same difficulty arises for $C^{s+\alpha}$ [[nonlocal Evans-Krylov theorem|estimates for convex equations]]. For example, is it true that a bounded function $u$ such that $M^+u = 0$ in $B_1$, where $M^+$ is the [[extremal operators|monster Pucci operator]] is $C^{s+\alpha}$ for some $\alpha>0$?
| | This definition is important for applications to probability. |
|
| |
|
| == A nonlocal generalization of the parabolic [[Krylov-Safonov theorem]] == | | == 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. |
|
| |
|
| Let $u$ be a bounded function in $\R^n \times [-1,0]$ such that it solves an integro-differential parabolic equation
| | == Heat kernel == |
| \[ u_t - \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y = 0 \qquad \text{in } B_1 \times (-1,0).\]
| | The fractional heat kernel is the function $p(t,x)$ which solves the equation |
| Making the usual symmetry and uniform ellipticity assumptions on the kernel $K$:
| |
| \begin{align*} | | \begin{align*} |
| K(x,y) &= K(x,-y) \\
| | p(0,x) &= \delta_0 \\ |
| \frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(x,y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2).
| | p_t(t,x) + (-\Delta)^s p &= 0 |
| \end{align*} | | \end{align*} |
| Is it true that the solutions $u$ is Holder continuous in $B_{1/2} \times [-1/2,0]$, with an estimate
| |
| \[ ||u||_{C^\alpha(B_{1/2} \times [-1/2,0])} \leq C ||u||_{L^\infty(\R^n \times [-1,0])}, \]
| |
| for constants $C$ and $\alpha>0$ which do not blow up as $s \to 2$?
| |
|
| |
| For an estimate with constants that blow up as $s \to 2$, one can easily adapt an argument for [[drift-diffusion equations]] <ref name="S2"/>.
| |
|
| |
|
| The elliptic version of this result is well known <ref name="CS"/>. The proof is not easy to adapt to the parabolic case because the [[Alexadroff-Bakelman-Pucci estimate]] is quite different in the elliptic and parabolic case. | | 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 $C$ |
| | \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \] |
|
| |
|
| For gradient flows of Dirichlet forms, the problems appears open as well. However, it is conceivable that one could adapt the proof of the stationary case <ref name="K"/> to obtain the result without a major difficulty.
| | Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling |
| | | \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \] |
| == Optimal regularity for the [[obstacle problem]] for a general integro-differential operator ==
| |
| | |
| Let $u$ be the solution to the [[obstacle problem for the fractional laplacian]],
| |
| \begin{align*}
| |
| u &\geq \varphi \qquad \text{in } \R^n, \\
| |
| (-\Delta)^{s/2} u &\geq 0 \qquad \text{in } \R^n, \\
| |
| (-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\
| |
| \end{align*}
| |
| where $\varphi$ is a smooth compactly supported function. It is known that $u \in C^{1,s/2}$ (where $s$ coincides with the order of the fractional Laplacian). This regularity is optimal in the sense that one can construct solutions that are not in $C^{1,s/2+\varepsilon}$ for any $\varepsilon>0$. One can consider the same problem replacing the fractional Laplacian by any other nonlocal operator. In fact, this problem corresponds to the [[optimal stopping problem]] in stochastic control, with applications to mathematical finance. The fractional Laplacian is just the particular case when the [[Levy process]] involved is $\alpha$-stable. The optimal regularity for the general problem is currently an open problem. Even in the linear case with constant coefficients this is nontrivial. If $u$ is a solution of
| |
| \begin{align*} | |
| u &\geq \varphi \qquad \text{in } \R^n, \\
| |
| L u &\leq 0 \qquad \text{in } \R^n, \\
| |
| L u &= 0 \qquad \text{in } \{u>\varphi\}, \\
| |
| \end{align*}
| |
| where $L$ is a [[linear integro-differential operator]], then what is the optimal regularity we can obtain for $u$?
| |
|
| |
|
| The optimal regularity would naturally depend on some assumptions on the linear operator $L$. If $L$ is a purely integro-differential with a kernel $K$ satisfying the usual ellipticity conditions
| | == 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*} | | \begin{align*} |
| K(y) &= K(-y) \\
| | u(x) &= g(x) \qquad \text{if } x \notin B_1 \\ |
| \frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2), | | (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1. |
| \end{align*} | | \end{align*} |
| it is natural to expect the solution $u$ to be $C^s$, but this regularity is not optimal. Is the optimal regularity going to be $C^{1,s/2}$ as in the fractional Laplacian case? Most probably some extra assumption on the kernel will be needed.
| |
|
| |
| A solution to this problem would be very interesting if it provides an optimal regularity result for a natural family of kernels. If the assumption is something hard to check (like for example that there exists an extension problem whose Dirichlet to Neumann map is $L$), then the result may not be that interesting.
| |
|
| |
| == Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==
| |
|
| |
| Consider a [[drift-diffusion equation]] of the form
| |
| \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]
| |
|
| |
|
| The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty(BMO)$ <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$). | | 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 |
| | \[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\] |
|
| |
|
| The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$. | | The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')<ref name="L"/>. |
|
| |
|
| == Complete classification of free boundary points in the [[fractional obstacle problem]] == | | == 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}}. |
|
| |
|
| Some free boundary points of the [[fractional obstacle problem]] are classified as regular and the free boundary is known to be smooth around them <ref name="CSS"/>. Other points on the free boundary are classified as singular, and they are shown to be contained in a lower dimensional differentiable surface, and therefore to be rare <ref name="GP"/>. However, there may be other points on the free boundary that do not fall under those two categories. Two questions need to be answered.\
| | === Full space regularization of the Riesz potential === |
| # Can there be any point on the free boundary that is neither regular nor singular? It is easy to produce examples in the [[thin obstacle problem]], using the [[extension technique]]. However, it is not clear if such examples can be made in the original formulation of the [[fractional obstacle problem]] since because of the decay at infinity requirement.
| | 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 $u \in B_{p,q}^{r+2s}$. |
| # In case that point of a third category exist, is the free boundary smooth around these points in the ''third category''?
| |
|
| |
|
| Other open problems concerning the [[fractional obstacle problem]] are
| | === Boundary regularity === |
| # Further regularity of the free boundary in smoother classes than $C^{1,\alpha}$.
| | 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). |
| # Regularity of the free boundary for the parabolic problem.
| |
|
| |
|
| == References == | | == References == |
| {{reflist|refs= | | {{reflist|refs= |
| <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="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="S2">{{Citation | last1=Silvestre | first1=Luis | title=Holder estimates for advection fractional-diffusion equations | year=To appear | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze}}</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> |
| <ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
| |
| <ref name="CV">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Vasseur | first2=Alexis | title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation | url=http://dx.doi.org/10.4007/annals.2010.171.1903 | doi=10.4007/annals.2010.171.1903 | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=3 | pages=1903–1930}}</ref>
| |
| <ref name="SSSZ">{{Citation | last1=Seregin | first1=G. | last2=Silvestre | first2=Luis | last3=Sverak | first3=V. | last4=Zlatos | first4=A. | title=On divergence-free drifts | year=2010 | journal=Arxiv preprint arXiv:1010.6025}}</ref>
| |
| <ref name="FV">{{Citation | last1=Friedlander | first1=S. | last2=Vicol | first2=V. | title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics | year=2011 | journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</ref>
| |
| <ref name="CRS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Roquejoffre | first2=Jean Michel |last3= Savin | first3= Ovidiu | title= Nonlocal Minimal Surfaces | url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract | doi=10.1002/cpa.20331 | year=2010 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0003-486X | volume=63 | issue=9 | pages=1111–1144}}</ref>
| |
| <ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</ref>
| |
| <ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
| |
| <ref name="GP">{{Citation | last1=Petrosyan | first1=A. | last2=Garofalo | first2=N. | title=Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2009 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=177 | issue=2 | pages=415–461}}</ref>
| |
| <ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-bakelman-pucci type estimates for integro-differential equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</ref>
| |
| <ref name="BI">{{Citation | last1=Barles | first1=Guy | last2=Imbert | first2=Cyril | title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited | url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 | doi=10.1016/j.anihpc.2007.02.007 | year=2008 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=25 | issue=3 | pages=567–585}}</ref>
| |
| }} | | }} |
The fractional Laplacian $-(-\Delta)^s$ $s>0$ 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 the operator enjoys a comparison principle, since if $u\leq v$ everywhere with equality a some point $x$ then $\delta u(x,y) \leq \delta v(x,y)$ for all $y$, implying that
\[ (-\Delta)^su(x) \geq (-\Delta)^sv(x) \]
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.
The operator can be defined as the generator of $\alpha$-stable Levy processes. More precisely, if $X_t$ is an $\alpha$-stable 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 is the function $p(t,x)$ 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
\[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]
The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')[1].
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 [2], 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
- ↑ Landkof, N. S. (1972), Foundations of modern potential theory, Berlin, New York: Springer-Verlag
- ↑ Silvestre, Luis (2007), "Regularity of the obstacle problem for a fractional power of the Laplace operator", Communications on Pure and Applied Mathematics 60 (1): 67–112, doi:10.1002/cpa.20153, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20153