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= Problems for integro-differential equations with rough coefficients or nonlinear equations =
Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.
== Hölder estimates for singular integro-differential equations ==


Consider an integro-differential equation of the form
== Elliptic case ==
\[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\]
=== Holder continuity ===
(An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin and not symmetric)
Given a bounded solution of the following elliptic PDE
 
\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]
[[Hölder estimates]] are known to hold under certain 'ellipticity' assumptions for the measures $\mu_x(y)$. In many cases, we consider the absolutely continuous version $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time the measures $\mu_x$ satisfy.
where repeated indices denotes summation and we assume
\[ \int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha}, \]
for all radius $R>0$ and $x \in B_1$, for some given constant $\alpha \in (0,2)$. This is the sharp assumption for stable operators that are independent of $x$ <ref name="ros2014regularity" />.
 
[[Hölder estimates]] are not known to hold under such generality. For the current methods, singular measures $\mu_x$ (without an absolutely continuous part) are out of reach. A new idea is needed in order to solve this problem.
 
Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$.
 
== An integral ABP estimate ==
 
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?
 
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*}
\begin{align*}
K(x,y) &= K(x,-y) \\
\lambda I &\leq \{a_{ij}(x)\} \leq \Lambda I \text{ for all $x$. (This is the uniform ellipticity condition)}\\
\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).
b &\in L^n(B_1), \\
f &\in L^n(B_1).
\end{align*}
\end{align*}
If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?
Then the function $u$ is Holder continuous and for some small $\alpha>0$ it satisfies the estimate
 
\[ ||u||_{C^\alpha(B_{1/2})} \leq C (||u||_{L^\infty(B_1)}+||f||_{L^n(B_1)}).\]
This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.<ref name="GS"/>
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.
 
== Holder estimates for parabolic equations with variable order ==
 
[[Holder estimates]] are known for elliptic and parabolic integro-differential equations with rough kernels. For elliptic equations, these estimates are available even when the order of the equation changes from point to point <ref name="BK"/> <ref name="S" />. Such estimate is not available for parabolic equations and it is not clear whether it holds.
 
More precisely, we would like to study a parabolic equation of the form
\[ u_t(t,x) = \int_{\R^n} (u(t,x+y) - u(t,x)) K(t,x,y) dy.\]
Here $K$ is symmetric (i.e. $K(t,x,y) = K(t,x,-y)$) and satisfies the bounds
\[ \frac \lambda {|y|^{n+s(t,x)}} \leq K(t,x,y) \leq \frac \Lambda {|y|^{n+s(t,x)}}.\]
The order of the equation $s(t,x) \in (0,1)$ changes from point to point and it should stay strictly away from zero. It would also make sense to study other families of [[linear integro-differential operators]]. Does a parabolic [[Holder estimate]] hold in this case?
 
== 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.<ref name="BI"/>
 
== Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==


Consider a [[drift-diffusion equation]] of the form
=== Harnack inequality ===
\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]
Given a nonnegative solution of the following elliptic PDE
\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]
Under the same assumptions as for the Holder estimates, the following Harnack inequality holds
\[ \sup_{B_{1/2}} u \leq C (\inf_{B_{1/2}} u+||f||_{L^n(B_1)}).\]
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.


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 $b \in L^\infty$ <ref name="SilHJ"/>, or $b \in L^\infty(BMO)$ and in addition $b$ is divergence free <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b$ is divergence free and $b \in L^\infty(BMO^{-1})$ (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 $b$ is divergence free and $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).
=== Viscosity solutions ===
 
Both the Holder estimates and the Harnack inequality can be applied to [[viscosity solutions]] of nonlinear equations. Formally, one can replace the equation (at least when $b=0$) by
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 $p$ if $\mathrm{div}\, b =0$. The case of arbitrary divergence might be more complicated.
 
= Open problems for equations related to fluids =
 
== Well posedness of the supercritical [[surface quasi-geostrophic equation]] and related problems ==
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*}
\theta(x,0) &= \theta_0(x) \\
M^+(D^2 u) &\geq f \text{ in } B_1,\\
\theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
M^-(D^2 u) &\leq f \text{ in } B_1.
\end{align*}
\end{align*}
where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
When $f$ is continuous, both inequalities above are well defined in the viscosity sense.


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 that the solution of this problem should be preceded by a better understanding of the inviscid problem (with the fractional diffusion term removed).


== Well posedness of the Hilbert flow problem ==
== Parabolic case ==
The elliptic case is implied by the results in the parabolic setting.


The Hilbert flow problem is a simple 1D toy model for fluid equations in higher dimensions. It was originally suggested in a paper by Cordoba, Cordoba and Fontelos.<ref name="cordobacordoba2005" /> The equation is in terms of a scalar function $\theta(t,x)$. Here $x \in \R$ is a one dimensional variable.
=== Holder estimates ===
\[ \theta_t + \mathrm H\theta \, \theta_x = 0.\]
The Holder estimate are similar in the parabolic case as in the elliptic case. Let us define the parabolic cylinder
The operator $\mathrm H$ stands for the Hilbert transform. There are several independent proofs that this equation develops singularities in finite time.<ref name="cordobacordoba2005" />
\[ Q_r(x_0,t_0) = \{(x,t) : |x-x_0|<r \text{ and } 0 \leq t_0 - t < t^2 \}.\]
<ref name="CCF2"/> <ref name="HDong"/> <ref name="K"/> <ref name="SV" /> The equation still develops singularities in finite time if we add fractional diffusion
\[ \theta_t + \mathrm H\theta \, \theta_x + (-\Delta)^s \theta = 0,\]
provided that $s < 1/4$.<ref name="HDong"/> <ref name="SV"/> <ref name="K"/> <ref name="li2011one" /> The equation is known to be classically well posed for $s \geq 1/2$. In the range $s \in [1/4,1/2)$, it is not known whether singularities may occur in finite time.


Silvestre and Vicol conjectured that the solution $\theta$ satisfies an a priori estimate in $C^{1/2}$ for positive time, both in the viscous and inviscid model.<ref name="SV" /> If this conjecture turns out to be true, the equation above will be well posed when $s > 1/4$.
Given a bounded solution of the parabolic PDE
 
\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]
= Open problems related to minimal surfaces and free boundaries =
where repeated indices denotes summation and we assume
 
== 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.
 
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
\[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]
 
When $s$ is sufficiently close to one, such set does not exist if $n < 8$.
 
== 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 and radially symmetric. 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*}
\begin{align*}
K(y) &= K(-y) \\
\lambda I &\leq \{a_{ij}(x,t)\} \leq \Lambda I \text{ for all $x$ and $t$. (This is the uniform ellipticity condition)}\\
\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),
b &\in L^n(Q_1), \\
f &\in L^n(Q_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.
Then the function $u$ is Holder continuous and satisfies the estimate
 
\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^n(Q_1)}).\]
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.
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.


UPDATE: This problem has been recently solved by Caffarelli, Ros-Oton, and Serra <ref name="CRS16" />.
===Harnack inequality ===
In the parabolic Harnack inequality, the infimum and the maximum must be taken in cylinders which are shifted in time.


== Complete understanding of free boundary points in the [[fractional obstacle problem]] ==
Given a nonnegative solution of the parabolic PDE
\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]
under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality
\[ \sup_{Q_{1/2}(0,0)} u \leq C \left(\inf_{Q_{1/2}(0,-1/2)} u+||f||_{L^n(Q_1)} \right).\]
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.


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 for $s=\frac12$ 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.\
== $C^{1,\alpha}$ estimates for fully nonlinear equations ==
# 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]] because of the decay at infinity requirement.
# In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?


UPDATE: It has been recently proved by Barrios, Figalli, and Ros-Oton <ref name="BFR"/> that, when the obstacle $\varphi$ satisfies $\Delta\varphi\leq0$, then regular and singular points do exhaust all free boundary points. Thus, under this ``concavity'' assumption, there are no free boundary points in the ``third category''.
The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain.
\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]
The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any vector $e$.


= References =
Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and
{{reflist|refs=
\[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\]
<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>
The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.
<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="CRS16">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Ros-Oton | first2=Xavier |last3= Serra | first3= Joaquim | title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries | year=2016 | journal=[[preprint arXiv (2016)]]}}</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>
<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="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="SV">{{Citation | last1=Silvestre | first1= Luis | last2=Vicol | first2= Vlad | title=On a transport equation with nonlocal drift | journal=arXiv preprint arXiv:1408.1056}}</ref>
<ref name="li2011one">{{Citation | last1=Li | first1= Dong | last2=Rodrigo | first2= José L | title=On a one-dimensional nonlocal flux with fractional dissipation | journal=SIAM Journal on Mathematical Analysis | year=2011 | volume=43 | pages=507--526}}</ref>
<ref name="cordobacordoba2005">{{Citation | last1=Córdoba | first1= Antonio | last2=Córdoba | first2= Diego | last3=Fontelos | first3= Marco A. | title=Formation of singularities for a transport equation with nonlocal velocity | url=http://dx.doi.org/10.4007/annals.2005.162.1377 | journal=Ann. of Math. (2) | issn=0003-486X | year=2005 | volume=162 | pages=1377--1389 | doi=10.4007/annals.2005.162.1377}}</ref>
<ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=J. Differential Equations, to appear.}}</ref>
<ref name="HDong">{{Citation | last1=Dong | first1= Hongjie | title=Well-posedness for a transport equation with nonlocal velocity | url=http://dx.doi.org/10.1016/j.jfa.2008.08.005 | journal=J. Funct. Anal. | issn=0022-1236 | year=2008 | volume=255 | pages=3070--3097 | doi=10.1016/j.jfa.2008.08.005}}</ref>
<ref name="K">{{Citation | last1=Kiselev | first1= A. | title=Regularity and blow up for active scalars | url=http://dx.doi.org/10.1051/mmnp/20105410 | journal=Math. Model. Nat. Phenom. | issn=0973-5348 | year=2010 | volume=5 | pages=225--255 | doi=10.1051/mmnp/20105410}}</ref>
<ref name="CCF2">{{Citation | last1=Córdoba | first1= Antonio | last2=Córdoba | first2= Diego | last3=Fontelos | first3= Marco A. | title=Integral inequalities for the Hilbert transform applied to a nonlocal transport equation | url=http://dx.doi.org/10.1016/j.matpur.2006.08.002 | journal=J. Math. Pures Appl. (9) | issn=0021-7824 | year=2006 | volume=86 | pages=529--540 | doi=10.1016/j.matpur.2006.08.002}}</ref>
<ref name="SilHJ">{{Citation | last1=Silvestre | first1= Luis | title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion | url=http://dx.doi.org/10.1016/j.aim.2010.09.007 | journal=Adv. Math. | issn=0001-8708 | year=2011 | volume=226 | pages=2020--2039 | doi=10.1016/j.aim.2010.09.007}}</ref>
}}

Revision as of 13:23, 23 February 2012

Krylov-Safonov theorem provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.

Elliptic case

Holder continuity

Given a bounded solution of the following elliptic PDE \[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\] where repeated indices denotes summation and we assume \begin{align*} \lambda I &\leq \{a_{ij}(x)\} \leq \Lambda I \text{ for all $x$. (This is the uniform ellipticity condition)}\\ b &\in L^n(B_1), \\ f &\in L^n(B_1). \end{align*} Then the function $u$ is Holder continuous and for some small $\alpha>0$ it satisfies the estimate \[ ||u||_{C^\alpha(B_{1/2})} \leq C (||u||_{L^\infty(B_1)}+||f||_{L^n(B_1)}).\] The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.

Harnack inequality

Given a nonnegative solution of the following elliptic PDE \[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\] Under the same assumptions as for the Holder estimates, the following Harnack inequality holds \[ \sup_{B_{1/2}} u \leq C (\inf_{B_{1/2}} u+||f||_{L^n(B_1)}).\] The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.

Viscosity solutions

Both the Holder estimates and the Harnack inequality can be applied to viscosity solutions of nonlinear equations. Formally, one can replace the equation (at least when $b=0$) by \begin{align*} M^+(D^2 u) &\geq f \text{ in } B_1,\\ M^-(D^2 u) &\leq f \text{ in } B_1. \end{align*} When $f$ is continuous, both inequalities above are well defined in the viscosity sense.


Parabolic case

The elliptic case is implied by the results in the parabolic setting.

Holder estimates

The Holder estimate are similar in the parabolic case as in the elliptic case. Let us define the parabolic cylinder \[ Q_r(x_0,t_0) = \{(x,t) : |x-x_0|<r \text{ and } 0 \leq t_0 - t < t^2 \}.\]

Given a bounded solution of the parabolic PDE \[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\] where repeated indices denotes summation and we assume \begin{align*} \lambda I &\leq \{a_{ij}(x,t)\} \leq \Lambda I \text{ for all $x$ and $t$. (This is the uniform ellipticity condition)}\\ b &\in L^n(Q_1), \\ f &\in L^n(Q_1). \end{align*} Then the function $u$ is Holder continuous and satisfies the estimate \[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^n(Q_1)}).\] The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.

Harnack inequality

In the parabolic Harnack inequality, the infimum and the maximum must be taken in cylinders which are shifted in time.

Given a nonnegative solution of the parabolic PDE \[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\] under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality \[ \sup_{Q_{1/2}(0,0)} u \leq C \left(\inf_{Q_{1/2}(0,-1/2)} u+||f||_{L^n(Q_1)} \right).\] The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.

$C^{1,\alpha}$ estimates for fully nonlinear equations

The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain. \[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \] The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any vector $e$.

Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and \[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\] The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.