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Holder continuity of the solutions can sometimes be proved only from ellipticity assumptions on the equation, without depending on smoothness of the coefficients. This allows great flexibility in terms of applications of the result. The corresponding result for elliptic equations of second order is the [[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser]] theorem in the divergence form.
== Well posedness of the supercritical [[surface quasi-geostrophic equation]] ==
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.


The holder estimates are closely related to the [[Harnack inequality]].
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.


There are integro-differential versions of both [[De Giorgi-Nash-Moser]] theorem and [[Krylov-Safonov]] theorem. The former uses variational techniques and is stated in terms of Dirichlet forms. The latter is based on comparison principles.
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 $s \geq 1/2$.


A Holder estimate says that a solution to an [[integro-differential equation]] $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$ (small). It is '''very important''' to allow for a very rough dependence of $L_x$ with respect to $x$, since the result then applies to the linearization of fully nonlinear equations without any extra a priori estimate. On the other hand, the linearization of a fully nonlinear equation (for example the [[Isaacs equation]]) would inherit the initial assumptions regarding for the kernels with respect to $y$. Therefore, smoothness (or even structural) assumptions for the kernels with respect to $y$ can be made keeping such result useful.
== Regularity of [[nonlocal minimal surfaces]] ==


In the non variational setting the integro-differential operators $L_x$ will be assumed to belong to some family, but no continuity is assumed for its dependence with respect to $x$. Typically, $L_x u(x)$ has the form
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.
$$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy$$
Since [[integro-differential equations]] allow for a great flexibility of equations, there are several variations on the result: different assumptions on the kernels, mixed local terms, evolution equations, etc. The linear equation with rough coefficients is equivalent to the function $u$ satisfying two inequalities for the [[extremal operators]] corresponding to the family of operators $L$, which stresses the nonlinear character of the estimates.


Some estimates blow up as the order of the equation converges to two, and others pass to the limit. In the latter case the estimates are a true generalization of either [[De Giorgi-Nash-Moser]] or [[Krylov-Safonov]]]
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
# $A$ 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
\[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]


== Estimates which blow up as the order goes to two ==
When $s$ is sufficiently close to one, such set does not exist if $n < 8$.


=== Non variational case ===
== An integral ABP estimate ==


The Holder estimates were first obtained using probabilistic techniques <ref name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to a family satisfying certain set of assumptions. No regularity of any kind is assumed for $K$ with respect to $x$. The assumption for the family of operators are
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?
# '''Scaling''': If $L$ belongs to the family, then so does its scaled version $L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
# '''Nondegeneracy''': If $K$ is the kernel associated to $L$, $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\sup_{B_1} K} \leq C_1$ for some $C_1$ and $\alpha>0$ independent of $K$.


The right hand side $f$ is assumed to belong to $L^\infty$.
Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
\[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
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 particular cases in which this result applies is the uniformly elliptic case.
This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.<ref name="GS"/>
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no continuity of $s$ respect to $x$ is required.
The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$. However this assumption can be overcome in the following two situations.
* For $s<1$, the symmetry assumption can be removed if the equation does not contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy = f(x)$ in $B_1$.
* For $s>1$, the symmetry assumption can be removed if the drift correction term is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy = f(x)$ in $B_1$.


The reason for the symmetry assumption, or the modification of the drift correction term, is that in the original formulation the term $y \cdot \nabla u(x) \, \chi_{B_1}(y)$ is not scale invariant.
== 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?


=== Variational case ===
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 $x$, e.g.
\begin{align*}
I[u] = \int (u(x+z) - u(x) - Du(x)\cdot z 1_{B}(z))\mu_x(dz)
\end{align*}
where $(\mu_x)_x$ is a family of L\'evy measures, H\"older continous with 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"/>


A [[Dirichlet forms]] is a quadratic functional of the form
== A local [[differentiability estimates|$C^{1,\alpha}$ estimate]] for integro-differential equations with nonsmooth kernels ==
$$ J(u) = \iint_{\R^n \times \R^n} |u(x)-u(y)|^2 K(x,y) \, dx \, dy $$.


Minimizers of Dirichlet forms are a nonlocal version of minimizers of integral functionals as in [[De Giorgi-Nash-Moser]] theorem.
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
\[ \frac{\lambda(2-s)}{|y|^{n+s}} \leq K(y) \leq \frac{\Lambda(2-s)}{|y|^{n+s}}. \]
Is it true that $u \in C^{1,\alpha}(B_1)$?


The symmetry assumption $K(x,y)=K(y,x)$ is natural since the skew-symmetric part of $K$ would be ignored by the quadratic functional.
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 $s>1$.


It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Holder continuous <ref name="CCV"/>. The method of the proof builds an integro-differential version of the parabolic De Giorgi technique that was developed for the study of critical [[surface quasi-geostrophic equation]].  
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.


At some point in the original proof of De Giorgi, it is used that the characteristic functions of a set of positive measure do not belong to $H^1$. Moreover, a quantitative estimate is required about the measure of ''intermediate'' level sets for $H^1$ functions. In the integro-differential context, the required statement to carry out the proof would be the same with the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would even require a non trivial proof for $s$ close to $2$. The difficulty is bypassed though an argument that takes advantage of the nonlocal character of the equation, similarly as in {{Citation needed}}.
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.


== Estimates which pass to the second order limit ==
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.


=== Non variational case ===
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 $\alpha>0$?


An integro-differential generalization of [[Krylov-Safonov]] theorem is available <ref name="CS"/>. The assumption on the kernels are
== A nonlocal generalization of the parabolic [[Krylov-Safonov theorem]] ==
# '''Symmetry''': $K(x,y) = K(x,-y)$.
# '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.


The right hand side $f$ is assumed to be in $L^\infty$. The constants in the Holder estimate do not blow up as $s \to 2$.
Let $u$ be a bounded function in $\R^n \times [-1,0]$ such that it solves an integro-differential parabolic equation
\[ u_t - \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y = 0 \qquad \text{in } B_1 \times (-1,0).\]
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 $s \to 2$?


=== Variational case ===
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"/>.


In the stationary case, it is known that minimizers of Dirichlet forms are Holder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser]] to the nonlocal setting <ref name="K"/>.
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.
 
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.
 
== 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
\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 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 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 $p$.
 
== Complete classification of free boundary points in the [[fractional obstacle problem]] ==
 
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.\
# 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.
# 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
# Further regularity of the free boundary in smoother classes than $C^{1,\alpha}$.
# Regularity of the free boundary for the parabolic problem.


== References ==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
<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="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="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</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="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="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="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Hölder continuity of harmonic functions with respect to operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 | doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=30 | issue=7 | pages=1249–1259}}</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="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</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>
}}
}}

Revision as of 21:44, 23 February 2012

Well posedness of the supercritical surface quasi-geostrophic equation

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 [1]. 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.

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

  1. $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
  2. 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 $n < 8$.

An integral ABP estimate

The nonlocal version of the Alexadroff-Bakelman-Pucci estimate holds either for a right hand side in $L^\infty$ [2] (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels [3]. Would the following result be true?

Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$, \[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \] 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?

This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.[3]

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 $x$, e.g. \begin{align*} I[u] = \int (u(x+z) - u(x) - Du(x)\cdot z 1_{B}(z))\mu_x(dz) \end{align*} where $(\mu_x)_x$ is a family of L\'evy measures, H\"older continous with respect to $x$?

Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.[4]

A local $C^{1,\alpha}$ estimate for integro-differential equations with nonsmooth kernels

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 \[ \frac{\lambda(2-s)}{|y|^{n+s}} \leq K(y) \leq \frac{\Lambda(2-s)}{|y|^{n+s}}. \] 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 $C^{1,\alpha}$ estimates [2]. It is conceivable that the assumption may not be necessary at least for $s>1$.

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.

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.

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.

The same difficulty arises for $C^{s+\alpha}$ 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 monster Pucci operator is $C^{s+\alpha}$ for some $\alpha>0$?

A nonlocal generalization of the parabolic Krylov-Safonov theorem

Let $u$ be a bounded function in $\R^n \times [-1,0]$ such that it solves an integro-differential parabolic equation \[ u_t - \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y = 0 \qquad \text{in } B_1 \times (-1,0).\] 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 [5].

The elliptic version of this result is well known [2]. 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.

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 [6] to obtain the result without a major difficulty.

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 \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 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 $b \in L^\infty(BMO)$ [7]. 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) [8] [9]. 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 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 [7] can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$.

Complete classification of free boundary points in the fractional obstacle problem

Some free boundary points of the fractional obstacle problem are classified as regular and the free boundary is known to be smooth around them [10]. 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 [11]. However, there may be other points on the free boundary that do not fall under those two categories. Two questions need to be answered.\

  1. 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.
  2. 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

  1. Further regularity of the free boundary in smoother classes than $C^{1,\alpha}$.
  2. Regularity of the free boundary for the parabolic problem.

References

  1. Caffarelli, Luis A.; Roquejoffre, Jean Michel; Savin, Ovidiu (2010), "Nonlocal Minimal Surfaces", Communications on Pure and Applied Mathematics 63 (9): 1111–1144, doi:10.1002/cpa.20331, ISSN 0003-486X, http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract 
  2. 2.0 2.1 2.2 Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 
  3. 3.0 3.1 Guillen, N.; Schwab, R. (2010), "Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations", Arxiv preprint arXiv:1101.0279  Cite error: Invalid <ref> tag; name "GS" defined multiple times with different content
  4. Barles, Guy; Imbert, Cyril (2008), "Second-order elliptic integro-differential equations: viscosity solutions' theory revisited", Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 25 (3): 567–585, doi:10.1016/j.anihpc.2007.02.007, ISSN 0294-1449, http://dx.doi.org/10.1016/j.anihpc.2007.02.007 
  5. Silvestre, Luis (To appear), "Holder estimates for advection fractional-diffusion equations", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze 
  6. Kassmann, Moritz (2009), "A priori estimates for integro-differential operators with measurable kernels", Calculus of Variations and Partial Differential Equations 34 (1): 1–21, doi:10.1007/s00526-008-0173-6, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-008-0173-6 
  7. 7.0 7.1 Caffarelli, Luis A.; Vasseur, Alexis (2010), "Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation", Annals of Mathematics. Second Series 171 (3): 1903–1930, doi:10.4007/annals.2010.171.1903, ISSN 0003-486X, http://dx.doi.org/10.4007/annals.2010.171.1903 
  8. Friedlander, S.; Vicol, V. (2011), "Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics", Annales de l'Institut Henri Poincare (C) Non Linear Analysis 
  9. Seregin, G.; Silvestre, Luis; Sverak, V.; Zlatos, A. (2010), "On divergence-free drifts", Arxiv preprint arXiv:1010.6025 
  10. 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 
  11. 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