Obstacle problem for the fractional Laplacian and De Giorgi-Nash-Moser theorem: Difference between pages

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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 $s \in (0,1)$. The equation reads
The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form.
\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]].
The equation is
\[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \]
in the elliptic case, or
\[ u_t = \mathrm{div} A(x,t) \nabla u(x). \]
Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$,
\[ \langle A v,v \rangle \geq \lambda |v|^2,\]
for every $v \in \R^n$, uniformly in space and time.


== Existence and uniqueness ==
The corresponding result in non divergence form is [[Krylov-Safonov theorem]].
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:
For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].
\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.
== Elliptic version ==
For the result in the elliptic case, we assume that the equation
\[ \mathrm{div} A(x) \nabla u(x) = 0 \]
is satisfied in the unit ball $B_1$ of $\R^n$.
===Holder estimate===
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and
\[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.


== Regularity considerations ==
The result can be scaled to balls of arbitrary radius $r>0$ to obtain
=== Regularity of the solution ===
\[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]
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.
Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$.


Below, we outline the steps leading to the optimal regularity with a sketch of the ideas used in the proofs.
===Harnack inequality===
The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$:
\[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\]
The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.


==== * Almost $C^{2s}$ regularity ====
===Minimizers of convex functionals===
This first step of the proof is the simplest and it is the only step which is an adaptation of the classical case $s=1$.  
The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals
\[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\]
are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by [[bootstrapping]] with the [[Schauder estimates]] and the smoothness of $F$.


From the statement of the equation we have $(-\Delta)^s u \geq 0$.
Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made.


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: $D^2 u \geq -C I$.
== Parabolic version ==
For the result in the parabolic case, we assume that the equation
\[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \]
is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.
===Holder estimate===
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and
\[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.


Interpolating the semiconvexity and $L^\infty$ boundedness of $u$, we obtain that $(-\Delta)^s u \leq C$ for some constant $C$.
===Harnack inequality===
The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time:
\[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]


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 $\alpha < s$.
===Gradient flows===
 
The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth.
==== * $C^{2s+\alpha}$ regularity, for some small $\alpha>0$ ====
\[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\]
Let $w(x) = (-\Delta)^s u(x)$. A key observation is that the function $w$ satisfies the equation
The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.
\begin{align}
(-\Delta)^{1-s} w = -\Delta \varphi \qquad \text{in } \{u=\varphi\}, \\
w = 0 \varphi \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 <ref name="S"/>.
 
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 $\alpha < 1-s$ though 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"/>.
 
=== 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"/>.
 
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 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 <ref name="CF"/>. 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 ==
{{reflist|refs=
<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="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="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/>
}}

Revision as of 15:14, 14 March 2012

The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form.

The equation is \[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \] in the elliptic case, or \[ u_t = \mathrm{div} A(x,t) \nabla u(x). \] Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$, \[ \langle A v,v \rangle \geq \lambda |v|^2,\] for every $v \in \R^n$, uniformly in space and time.

The corresponding result in non divergence form is Krylov-Safonov theorem.

For nonlocal equations, there are analogous results both for Holder estimates and the Harnack inequality.

Elliptic version

For the result in the elliptic case, we assume that the equation \[ \mathrm{div} A(x) \nabla u(x) = 0 \] is satisfied in the unit ball $B_1$ of $\R^n$.

Holder estimate

The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and \[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\] The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.

The result can be scaled to balls of arbitrary radius $r>0$ to obtain \[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]

Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$.

Harnack inequality

The Harnack inequality says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$: \[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\] The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.

Minimizers of convex functionals

The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals \[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\] are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by bootstrapping with the Schauder estimates and the smoothness of $F$.

Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made.

Parabolic version

For the result in the parabolic case, we assume that the equation \[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \] is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.

Holder estimate

The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and \[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\] The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.

Harnack inequality

The Harnack inequality says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time: \[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]

Gradient flows

The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth. \[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\] The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.