Open problems and Obstacle problem for the fractional Laplacian: Difference between pages

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== Well posedness of the supercritical [[surface quasi-geostrophic equation]] ==
The obstacle problem for the fractional Laplacian refers to the particular case of the [[obstacle problem]] when the elliptic operator $L$ is given by the [[fractional Laplacian]]: $L = -(-\Delta)^s$ for some $s \in (0,1)$. Given some smooth function $\varphi$, the equation reads
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}
\begin{align*}
u &\geq \varphi \qquad \text{everywhere}\\
\theta(x,0) &= \theta_0(x) \\
(-\Delta)^s u &\geq 0 \qquad \text{everywhere}\\
\theta_t + u \cdot \nabla \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
(-\Delta)^s u &= 0 \qquad \text{wherever } u > \varphi.
\end{align*}
\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.
The equation is derived from an [[optimal stopping problem]] when considering $\alpha$-stable Levy processes. It serves as the simplest model for other optimal stopping problems with purely jump processes and therefore its understanding is relevant for applications to [[financial mathematics]].


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 $s \geq 1/2$.
== Existence and uniqueness ==
The equation can be studied from either a variational or a non-variational point of view, and with or without boundary conditions.


== Regularity of [[nonlocal minimal surfaces]] ==
As a variational inequality the equation emerges as the minimizer of the homoegeneous $\dot H^s$ norm from all functions $u$ such that $u \geq \varphi$. In the case when the domain is the full space $\mathbb R^d$, a decay at infinity $u(x) \to 0$ as $|x| \to \infty$ is usually assumed. Note that in low dimensions $\dot H^s$ is not embedded in $L^p$ for any $p<\infty$ and therefore the boundary condition at infinity cannot be assured. In low dimensions one can overcome this inconvenience by minimizing the full $H^s$ norm and therefore obtaining the equation with an extra term of zeroth order:
\begin{align}
u &\geq \varphi \qquad \text{everywhere}\\
(-\Delta)^s u + u &\geq 0 \qquad \text{everywhere}\\
(-\Delta)^s u + u &= 0 \qquad \text{wherever } u > \varphi.
\end{align}
This extra zeroth order term does not affect any regularity consideration for the solution.


A nonlocal minimal surface that is sufficiently flat is known to be smooth. 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.
From a non variational point of view, the solution $u$ can be obtained as the smallest $s$-superharmonic function (i.e. $(-\Delta)^s u \geq 0$ such that $u \geq \varphi$. In low dimensions one cannot assure the boundary condition at infinity because of the impossibility of constructing barriers (this is related to the fact that the fundamental solutions $|x|^{-n+2s}$ fail to decay to zero at infinity if $2s \geq n$). This can be overcome with the addition of the zeroth order term or by the study of the problem in a bounded domain with Dirichlet boundary conditions in the complement.


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
== Regularity considerations ==
# $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
=== Regularity of the solution ===
# If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds
Assuming that the obstacle $\varphi$ is smooth, the optimal regularity of the solution is $C^{1,s}$.
\[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]


When $s$ is sufficiently close to one, such set does not exist if $n < 8$.
The regularity $C^{1,s}$ coincides with $C^{1,1}$ when $s=1$, which is the optimal regularity in the classical case of the Laplacian. However, adapting the ideas of the classical proof to the fractional case suggests that the optimal regularity should be only $C^{2s}$. The optimal regularity in the case $s<1$ is better than the order of the equation and cannot be justified by any simple scaling argument.


== An integral ABP estimate ==
<div style="background:#EEEEEE;">
'''Outline of the proof.'''


The nonlocal version of the [[Alexadroff-Bakelman-Pucci estimate]] holds either for a right hand side in $L^\infty$ (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels. Would the following result be true?
The proof consists of the following steps that we sketch below.


Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
*'''Almost $C^{2s}$ regularity'''
\[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
This first step of the proof is the simplest and it is the only step which is an adaptation of the classical case $s=1$.
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
\begin{align*}
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?


== A local [[differentiability estimates|$C^{1,\alpha}$ estimate]] for integro-differential equations with nonsmooth kernels ==
From the statement of the equation we have $(-\Delta)^s u \geq 0$.


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
Since the average of two $s$-superharmonic function is also $s$-superharmonic, one can see that for any $h \in \mathbb R^d$, the function $v(x):=(u(x+h)+u(x-h))/2 + C|h|^2$ is $s$ superharmonic and $v \geq \varphi$ if $C = ||D^2 \varphi||_{L^\infty}$. By the comparison principle $v \geq u$. This means that $u$ is semiconvex: $D^2 u \geq -C I$.
\[ \frac{\lambda(2-s)}{|y|^{n+s}} \leq K(y) \leq \frac{\Lambda(2-s)}{|y|^{n+s}}. \]
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]]. It is conceivable that the assumption may not be necessary at least for $s>1$.
Interpolating the semiconvexity and $L^\infty$ boundedness of $u$, we obtain that $(-\Delta)^s u \leq C$ for some constant $C$.


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.
The boundedness of $(-\Delta)^s u$ does not imply that $u \in C^{2s}$ but it does imply that $u \in C^\alpha$ for all $\alpha < s$.


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 $f \in C^\alpha$ would easily have $C^{1,\alpha}$ estimates without any smoothness restriction of the tails of the kernel.
*'''$C^{2s+\alpha}$ regularity, for some small $\alpha>0$'''
Let $w(x) = (-\Delta)^s u(x)$. A key observation is that the function $w$ satisfies the equation
\begin{align}
(-\Delta)^{1-s} w &= -\Delta \varphi \qquad \text{in } \{u=\varphi\}, \\
w &= 0 \qquad \text{outside } \{u=\varphi\}.
\end{align}


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.
This is a Dirichlet problem for the conjugate fractional Laplacian. However there are two difficulties. First of all we need to prove that $w$ is continuous on the boundary $\partial \{u=\varphi\}$. Second, this boundary can be highly irregular a priori so we cannot expect to obtain any H\"older continuity of $w$ from the Dirichlet problem alone.


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 $\alpha>0$?
From the semiconvexity of $u$ we have $-\Delta u \leq C$, and therefore we derive the extra condition $(-\Delta)^{1-s} w \leq C$ in the full space $\R^d$ (in particular across the boundary $\partial \{u=\varphi\}$). Moreover, we also know that $w \geq 0$ everywhere.


== A nonlocal generalization of the parabolic [[Krylov-Safonov theorem]] ==
The $C^\alpha$ Holder continuity of $w$ on the boundary $\partial \{u=\varphi\}$ is obtained from an [[iterative improvement of oscillation]] procedure. Since $w \geq 0$ and $(-\Delta)^{1-s} u \leq C$, for any $x_0$ on $\partial \{u=\varphi\}$ we can show that $\max_{B_r(x_0)} w$ decays provided that $\{u > \varphi\} \cap B_r$ is sufficiently "thick" using the [[weak Harnack inequality]]. We cannot rule out the case in which $\{u > \varphi\} \cap B_r$ has a very small measure. However, in the case that $\{u > \varphi\} \cap B_r$ is too small in measure, we can prove that $u$ separates very slowly from $\varphi$. This slow separation is used to prove that $w$ must also improve its oscillation and this step is particularly tricky <ref name="S"/>.


Let $u$ be a bounded function in $\R^n \times [-1,0]$ such that it solves an integro-differential parabolic equation
Once we know that $w(x) = (-\Delta)^s u(x)$ is $C^\alpha$, this implies that $u \in C^{2s+\alpha}$ by classical potential analysis theory.
\[ u_t - \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y = 0 \qquad \text{in } B_1 \times (-1,0).\]
Making the usual symmetry and uniform ellipticity assumptions on the kernel $K$:
\begin{align*}
K(x,y) &= K(x,-y) \\
\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).
\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"/>.
*'''$C^{1,s}$ regularity'''
If the contact set $\{u=\varphi\}$ is convex or at least has an exterior ball condition, a fairly simple barrier function can be constructed to show that $w$ must be $C^{1-s}$ on the boundary $\partial \{u=\varphi\}$. This is the generic boundary regularity for solutions of fractional Laplace equations in smooth domains.


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.
Without assuming anything on the contact set $\{u=\varphi\}$, one can still obtain that $w \in C^\alpha$ for every $\alpha < 1-s$ through an iterative use of barrier functions <ref name="S"/>. The sharp $w \in C^{1-s}$ regularity in full generality was obtained rewriting the equation as a [[thin obstacle problem]] using the [[extension technique]] and then applying blowup techniques, the Almgren monotonicity formula and classification of global solutions <ref name="CSS"/>.


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.
The $C^{1-s}$ regularity of $w$ implies that $u \in C^{1,s}$ by classical potential analysis.
</div>


== Optimal regularity for the obstacle problem for a general integro-differential operator ==
=== Regularity of the free boundary ===
A regular point of the free boundary is where the solution $u$ is exactly $C^{1,s}$ an no better. This is classified explicitly in terms of the limits of the Almgren frequency formula <ref name="CSS"/>. Around any regular point, the free boundary is a smooth $C^{1,\alpha}$ surface <ref name="CSS"/>.


Let $u$ be the solution to the [[obstacle problem for the fractional laplacian]],
A singular point is defined as a point on the free boundary where the measure of the contact set has vanishing density. More precisely, if
\begin{align*}
\[ \lim_{r \to 0} \frac{|\{u=\varphi\} \cap B_r|}{r^n} = 0.\]
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
In the case $s=1/2$, it was shown by Nicola Garofalo and Arshak Petrosyan <ref name="GP"/> that the singular points of the free boundary are contained  inside a differentiable surface. The proof is done in the context of the [[thin obstacle problem]] and presumably can be extended to other powers of the Laplacian using the [[extension technique]].
\begin{align*}
K(y) &= K(-y) \\
\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),
\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 problems whose Dirichlet to Neumann map is $L$), then the result may not be that interesting.
It is important to notice that the definitions of regular and singular points of the free boundary are mutually exclusive but they do not exhaust all possible free boundary points. It is an interesting open problem to understand what other type of free boundary points are possible if any.


== Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==
== The parabolic version ==


Consider a [[drift-diffusion equation]] of the form
The parabolic version of the fractional obstacle problem was studied by Caffarelli and Figalli <ref name="CF"/>. They concluded that the solution $u$ has the following regularity estimates.
\[ u_t = b \cdot \nabla u + (-\Delta)^s u = 0.\]
\begin{align}
u_t, (-\Delta)^s u \in LogLip_t C_x^{1-s}, \text{ if } s\leq 1/3,\\
u_t, (-\Delta)^s u \in C_{t,x}^{\frac{1-s}{2s},{1-s}}, \text{ if } s > 1/3.
\end{align}


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 applis for $s \in (1/2,1)$ if $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).
It turns out that it is crucial to consider solutions $u$ to be non decreasing in time (which is assured by taking the initial value coinciding with the obstacle). Otherwise the regularity of the solution is reduced to merely $C^{2s}$ in space.


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.
The regularity of the free boundary has not been explored in the parabolic setting yet.


== 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="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | publisher=Wiley Online Library | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</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="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="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="CF">{{Citation | last1=Figalli | first1=A. | last2=Caffarelli | first2=Luis | title=Regularity of solutions to the parabolic fractional obstacle problem | year=2011 | journal=Arxiv preprint arXiv:1101.5170}}</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="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="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>
}}
}}

Revision as of 12:28, 29 January 2012

The obstacle problem for the fractional Laplacian refers to the particular case of the obstacle problem when the elliptic operator $L$ is given by the fractional Laplacian: $L = -(-\Delta)^s$ for some $s \in (0,1)$. Given some smooth function $\varphi$, the equation reads \begin{align} u &\geq \varphi \qquad \text{everywhere}\\ (-\Delta)^s u &\geq 0 \qquad \text{everywhere}\\ (-\Delta)^s u &= 0 \qquad \text{wherever } u > \varphi. \end{align}

The equation is derived from an optimal stopping problem when considering $\alpha$-stable Levy processes. It serves as the simplest model for other optimal stopping problems with purely jump processes and therefore its understanding is relevant for applications to financial mathematics.

Existence and uniqueness

The equation can be studied from either a variational or a non-variational point of view, and with or without boundary conditions.

As a variational inequality the equation emerges as the minimizer of the homoegeneous $\dot H^s$ norm from all functions $u$ such that $u \geq \varphi$. In the case when the domain is the full space $\mathbb R^d$, a decay at infinity $u(x) \to 0$ as $|x| \to \infty$ is usually assumed. Note that in low dimensions $\dot H^s$ is not embedded in $L^p$ for any $p<\infty$ and therefore the boundary condition at infinity cannot be assured. In low dimensions one can overcome this inconvenience by minimizing the full $H^s$ norm and therefore obtaining the equation with an extra term of zeroth order: \begin{align} u &\geq \varphi \qquad \text{everywhere}\\ (-\Delta)^s u + u &\geq 0 \qquad \text{everywhere}\\ (-\Delta)^s u + u &= 0 \qquad \text{wherever } u > \varphi. \end{align} This extra zeroth order term does not affect any regularity consideration for the solution.

From a non variational point of view, the solution $u$ can be obtained as the smallest $s$-superharmonic function (i.e. $(-\Delta)^s u \geq 0$ such that $u \geq \varphi$. In low dimensions one cannot assure the boundary condition at infinity because of the impossibility of constructing barriers (this is related to the fact that the fundamental solutions $|x|^{-n+2s}$ fail to decay to zero at infinity if $2s \geq n$). This can be overcome with the addition of the zeroth order term or by the study of the problem in a bounded domain with Dirichlet boundary conditions in the complement.

Regularity considerations

Regularity of the solution

Assuming that the obstacle $\varphi$ is smooth, the optimal regularity of the solution is $C^{1,s}$.

The regularity $C^{1,s}$ coincides with $C^{1,1}$ when $s=1$, which is the optimal regularity in the classical case of the Laplacian. However, adapting the ideas of the classical proof to the fractional case suggests that the optimal regularity should be only $C^{2s}$. The optimal regularity in the case $s<1$ is better than the order of the equation and cannot be justified by any simple scaling argument.

Outline of the proof.

The proof consists of the following steps that we sketch below.

  • Almost $C^{2s}$ regularity

This first step of the proof is the simplest and it is the only step which is an adaptation of the classical case $s=1$.

From the statement of the equation we have $(-\Delta)^s u \geq 0$.

Since the average of two $s$-superharmonic function is also $s$-superharmonic, one can see that for any $h \in \mathbb R^d$, the function $v(x):=(u(x+h)+u(x-h))/2 + C|h|^2$ is $s$ superharmonic and $v \geq \varphi$ if $C = ||D^2 \varphi||_{L^\infty}$. By the comparison principle $v \geq u$. This means that $u$ is semiconvex: $D^2 u \geq -C I$.

Interpolating the semiconvexity and $L^\infty$ boundedness of $u$, we obtain that $(-\Delta)^s u \leq C$ for some constant $C$.

The boundedness of $(-\Delta)^s u$ does not imply that $u \in C^{2s}$ but it does imply that $u \in C^\alpha$ for all $\alpha < s$.

  • $C^{2s+\alpha}$ regularity, for some small $\alpha>0$

Let $w(x) = (-\Delta)^s u(x)$. A key observation is that the function $w$ satisfies the equation \begin{align} (-\Delta)^{1-s} w &= -\Delta \varphi \qquad \text{in } \{u=\varphi\}, \\ w &= 0 \qquad \text{outside } \{u=\varphi\}. \end{align}

This is a Dirichlet problem for the conjugate fractional Laplacian. However there are two difficulties. First of all we need to prove that $w$ is continuous on the boundary $\partial \{u=\varphi\}$. Second, this boundary can be highly irregular a priori so we cannot expect to obtain any H\"older continuity of $w$ from the Dirichlet problem alone.

From the semiconvexity of $u$ we have $-\Delta u \leq C$, and therefore we derive the extra condition $(-\Delta)^{1-s} w \leq C$ in the full space $\R^d$ (in particular across the boundary $\partial \{u=\varphi\}$). Moreover, we also know that $w \geq 0$ everywhere.

The $C^\alpha$ Holder continuity of $w$ on the boundary $\partial \{u=\varphi\}$ is obtained from an iterative improvement of oscillation procedure. Since $w \geq 0$ and $(-\Delta)^{1-s} u \leq C$, for any $x_0$ on $\partial \{u=\varphi\}$ we can show that $\max_{B_r(x_0)} w$ decays provided that $\{u > \varphi\} \cap B_r$ is sufficiently "thick" using the weak Harnack inequality. We cannot rule out the case in which $\{u > \varphi\} \cap B_r$ has a very small measure. However, in the case that $\{u > \varphi\} \cap B_r$ is too small in measure, we can prove that $u$ separates very slowly from $\varphi$. This slow separation is used to prove that $w$ must also improve its oscillation and this step is particularly tricky [1].

Once we know that $w(x) = (-\Delta)^s u(x)$ is $C^\alpha$, this implies that $u \in C^{2s+\alpha}$ by classical potential analysis theory.

  • $C^{1,s}$ regularity

If the contact set $\{u=\varphi\}$ is convex or at least has an exterior ball condition, a fairly simple barrier function can be constructed to show that $w$ must be $C^{1-s}$ on the boundary $\partial \{u=\varphi\}$. This is the generic boundary regularity for solutions of fractional Laplace equations in smooth domains.

Without assuming anything on the contact set $\{u=\varphi\}$, one can still obtain that $w \in C^\alpha$ for every $\alpha < 1-s$ through an iterative use of barrier functions [1]. The sharp $w \in C^{1-s}$ regularity in full generality was obtained rewriting the equation as a thin obstacle problem using the extension technique and then applying blowup techniques, the Almgren monotonicity formula and classification of global solutions [2].

The $C^{1-s}$ regularity of $w$ implies that $u \in C^{1,s}$ by classical potential analysis.

Regularity of the free boundary

A regular point of the free boundary is where the solution $u$ is exactly $C^{1,s}$ an no better. This is classified explicitly in terms of the limits of the Almgren frequency formula [2]. Around any regular point, the free boundary is a smooth $C^{1,\alpha}$ surface [2].

A singular point is defined as a point on the free boundary where the measure of the contact set has vanishing density. More precisely, if \[ \lim_{r \to 0} \frac{|\{u=\varphi\} \cap B_r|}{r^n} = 0.\]

In the case $s=1/2$, it was shown by Nicola Garofalo and Arshak Petrosyan [3] that the singular points of the free boundary are contained inside a differentiable surface. The proof is done in the context of the thin obstacle problem and presumably can be extended to other powers of the Laplacian using the extension technique.

It is important to notice that the definitions of regular and singular points of the free boundary are mutually exclusive but they do not exhaust all possible free boundary points. It is an interesting open problem to understand what other type of free boundary points are possible if any.

The parabolic version

The parabolic version of the fractional obstacle problem was studied by Caffarelli and Figalli [4]. They concluded that the solution $u$ has the following regularity estimates. \begin{align} u_t, (-\Delta)^s u \in LogLip_t C_x^{1-s}, \text{ if } s\leq 1/3,\\ u_t, (-\Delta)^s u \in C_{t,x}^{\frac{1-s}{2s},{1-s}}, \text{ if } s > 1/3. \end{align}

It turns out that it is crucial to consider solutions $u$ to be non decreasing in time (which is assured by taking the initial value coinciding with the obstacle). Otherwise the regularity of the solution is reduced to merely $C^{2s}$ in space.

The regularity of the free boundary has not been explored in the parabolic setting yet.

References

  1. ↑ 1.0 1.1 Silvestre, Luis (2007), "Regularity of the obstacle problem for a fractional power of the Laplace operator", Communications on Pure and Applied Mathematics (Wiley Online Library) 60 (1): 67–112, ISSN 0010-3640 
  2. ↑ 2.0 2.1 2.2 Caffarelli, Luis A.; Salsa, Sandro; Silvestre, Luis (2008), "Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian", Inventiones Mathematicae 171 (2): 425–461, doi:10.1007/s00222-007-0086-6, ISSN 0020-9910, http://dx.doi.org/10.1007/s00222-007-0086-6 
  3. ↑ Petrosyan, A.; Garofalo, N. (2009), "Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem", Inventiones Mathematicae (Berlin, New York: Springer-Verlag) 177 (2): 415–461, ISSN 0020-9910 
  4. ↑ Figalli, A.; Caffarelli, Luis (2011), "Regularity of solutions to the parabolic fractional obstacle problem", Arxiv preprint arXiv:1101.5170 
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