# Open problems

(Difference between revisions)
 Revision as of 00:29, 6 March 2012 (view source)Luis (Talk | contribs)m (→Complete classification of free boundary points in the fractional obstacle problem)← Older edit Latest revision as of 14:48, 26 March 2016 (view source)Xavi (Talk | contribs) m (→Complete understanding of free boundary points in the fractional obstacle problem) (42 intermediate revisions not shown) Line 1: Line 1: - == Well posedness of the supercritical [[surface quasi-geostrophic equation]] == + = Problems for integro-differential equations with rough coefficients or nonlinear equations = - 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? + == Hölder estimates for singular integro-differential equations == - \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. + Consider an integro-differential equation of the form + $\int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.$ + (An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin and not symmetric) - 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 $s \geq 1/2$. + [[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. + $\int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha},$ + 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 $x$ . - == Regularity of [[nonlocal minimal surfaces]] == + [[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. - 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. + Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$. - + - 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 $n < 8$. + == An integral ABP estimate == == An integral ABP estimate == Line 36: Line 28: This type of estimate is currently known only under strong structural hypothesis on the kernels $K$. This type of estimate is currently known only under strong structural hypothesis on the kernels $K$. + + == 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 . 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 == == A [[comparison principle]] for $x$-dependent nonlocal equations which are '''not''' in the Levy-Ito form == Line 53: Line 55: Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form. Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form. - == A local [[differentiability estimates|$C^{1,\alpha}$ estimate]] for integro-differential equations with nonsmooth kernels == + == Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) == - 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 + Consider a [[drift-diffusion equation]] of the form - $\frac{\lambda(2-s)}{|y|^{n+s}} \leq K(y) \leq \frac{\Lambda(2-s)}{|y|^{n+s}}.$ + $u_t + b \cdot \nabla u + (-\Delta)^s u = 0.$ - 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 $s>1$. + 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 $b \in L^\infty$ , or $b \in L^\infty(BMO)$ and in addition $b$ is divergence free . 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) . 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 $(-\Delta)^{1-2s} b \in L^\infty(BMO)$). - 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 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 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. - 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. + = Open problems for equations related to fluids = - 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. + == 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*} + \theta(x,0) &= \theta_0(x) \\ + \theta_t + u \cdot \nabla \theta + (-\Delta)^s \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 same difficulty arises for $C^{s+\alpha}$ [[nonlocal Evans-Krylov theorem|estimates for convex equations]]. For example, is it true that a bounded function $u$ such that $M^+u = 0$ in $B_1$, where $M^+$ is the [[extremal operators|monster Pucci operator]] is $C^{s+\alpha}$ for some $\alpha>0$? + This 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 == + + 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. The equation is in terms of a scalar function $\theta(t,x)$. Here $x \in \R$ is a one dimensional variable. + $\theta_t + \mathrm H\theta \, \theta_x = 0.$ + The operator $\mathrm H$ stands for the Hilbert transform. There are several independent proofs that this equation develops singularities in finite time. + 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 $s < 1/4$. 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. If this conjecture turns out to be true, the equation above will be well posed when $s > 1/4$. + + = Open problems related to minimal surfaces and free boundaries = + + == Regularity of [[nonlocal minimal surfaces]] == + + 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. + + 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 == == Optimal regularity for the [[obstacle problem]] for a general integro-differential operator == Line 77: Line 108: (-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\ (-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\ \end{align*} \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 + 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*} \begin{align*} u &\geq \varphi \qquad \text{in } \R^n, \\ u &\geq \varphi \qquad \text{in } \R^n, \\ Line 94: Line 125: 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. 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 $s>1/2$) == + UPDATE: This problem has been recently solved by Caffarelli, Ros-Oton, and Serra . - + - 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 $b \in L^\infty(BMO)$ . 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) . 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 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]] == + == Complete understanding 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 . 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 . However, there may be other points on the free boundary that do not fall under those two categories. Two questions need to be answered.\ + Some free boundary points of the [[fractional obstacle problem]] are classified as regular and the free boundary is known to be smooth around them . 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 . 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]] because of the decay at infinity requirement. # 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''? # In case that a 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 + UPDATE: It has been recently proved by Barrios, Figalli, and Ros-Oton 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''. - # 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= {{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}} {{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}} Line 120: Line 142: {{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}} {{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}} {{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}} {{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}} + {{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)]]}} {{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}} {{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}} {{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}} {{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}} Line 125: Line 148: {{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}} {{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}} {{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}} {{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}} + {{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}} + {{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}} + {{Citation | last1=Silvestre | first1= Luis | last2=Vicol | first2= Vlad | title=On a transport equation with nonlocal drift | journal=arXiv preprint arXiv:1408.1056}} + {{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}} + {{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}} + {{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=J. Differential Equations, to appear.}} + {{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}} + {{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}} + {{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}} + {{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}} + {{Citation | last1=Barrios | first1= B. | last2=Figalli | first2= A. | last3=Ros-Oton | first3= X. | title=Global regularity for the free boundary in the obstacle problem for the fractional Laplacian | journal=preprint arXiv}} }} }}

# Problems for integro-differential equations with rough coefficients or nonlinear equations

## Hölder estimates for singular integro-differential equations

Consider an integro-differential equation of the form $\int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.$ (An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin and not symmetric)

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. $\int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha},$ 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 $x$ [1].

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 $L^\infty$ [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]

## 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 [4] [5]. 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.[6]

## 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 $b \in L^\infty$ [7], or $b \in L^\infty(BMO)$ and in addition $b$ is divergence free [8]. 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) [9] [10]. 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 $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).

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 [8] 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*} \theta(x,0) &= \theta_0(x) \\ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \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 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

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.[11] The equation is in terms of a scalar function $\theta(t,x)$. Here $x \in \R$ is a one dimensional variable. $\theta_t + \mathrm H\theta \, \theta_x = 0.$ The operator $\mathrm H$ stands for the Hilbert transform. There are several independent proofs that this equation develops singularities in finite time.[11] [12] [13] [14] [15] 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 $s < 1/4$.[13] [15] [14] [16] 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.[15] If this conjecture turns out to be true, the equation above will be well posed when $s > 1/4$.

# Open problems related to minimal surfaces and free boundaries

## Regularity of nonlocal minimal surfaces

A nonlocal minimal surface that is sufficiently flat is known to be smooth [17]. 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 $n < 8$.

## 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*} 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.

UPDATE: This problem has been recently solved by Caffarelli, Ros-Oton, and Serra [18].

## Complete understanding 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 [19]. 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 [20]. 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 because of the decay at infinity requirement.
2. 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 [21] 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.

# References

1. Ros-Oton, Xavier; Serra, Joaquim, "Regularity theory for general stable operators", J. Differential Equations, to appear.
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
3. 3.0 3.1 Guillen, N.; Schwab, R. (2010), "Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations", Arxiv preprint arXiv:1101.0279
4. Bass, Richard F.; Kassmann, Moritz (2005), [http://dx.doi.org/10.1080/03605300500257677 "Hölder continuity of harmonic functions with respect to operators of variable order"], Communications in Partial Differential Equations 30 (7): 1249–1259, doi:10.1080/03605300500257677, ISSN 0360-5302
5. Silvestre, Luis (2006), [http://dx.doi.org/10.1512/iumj.2006.55.2706 "Hölder estimates for solutions of integro-differential equations like the fractional Laplace"], Indiana University Mathematics Journal 55 (3): 1155–1174, doi:10.1512/iumj.2006.55.2706, ISSN 0022-2518
6. 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
7. Silvestre, Luis (2011), "On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion", Adv. Math. 226: 2020--2039, doi:10.1016/j.aim.2010.09.007, ISSN 0001-8708
8. 8.0 8.1 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
9. 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
10. Seregin, G.; Silvestre, Luis; Sverak, V.; Zlatos, A. (2010), "On divergence-free drifts", Arxiv preprint arXiv:1010.6025
11. 11.0 11.1 Córdoba, Antonio; Córdoba, Diego; Fontelos, Marco A. (2005), "Formation of singularities for a transport equation with nonlocal velocity", Ann. of Math. (2) 162: 1377--1389, doi:10.4007/annals.2005.162.1377, ISSN 0003-486X
12. Córdoba, Antonio; Córdoba, Diego; Fontelos, Marco A. (2006), "Integral inequalities for the Hilbert transform applied to a nonlocal transport equation", J. Math. Pures Appl. (9) 86: 529--540, doi:10.1016/j.matpur.2006.08.002, ISSN 0021-7824
13. 13.0 13.1 Dong, Hongjie (2008), "Well-posedness for a transport equation with nonlocal velocity", J. Funct. Anal. 255: 3070--3097, doi:10.1016/j.jfa.2008.08.005, ISSN 0022-1236
14. 14.0 14.1 Kiselev, A. (2010), "Regularity and blow up for active scalars", Math. Model. Nat. Phenom. 5: 225--255, doi:10.1051/mmnp/20105410, ISSN 0973-5348
15. 15.0 15.1 15.2 Silvestre, Luis; Vicol, Vlad, "On a transport equation with nonlocal drift", arXiv preprint arXiv:1408.1056
16. Li, Dong; Rodrigo, José L (2011), "On a one-dimensional nonlocal flux with fractional dissipation", SIAM Journal on Mathematical Analysis 43: 507--526
17. 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
18. Caffarelli, Luis A.; Ros-Oton, Xavier; Serra, Joaquim (2016), "Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries", preprint arXiv (2016)
19. 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
20. 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
21. Barrios, B.; Figalli, A.; Ros-Oton, X., "Global regularity for the free boundary in the obstacle problem for the fractional Laplacian", preprint arXiv