Open problems

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)

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 $x$ .

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 $L^\infty$ (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels . Would the following result be true?

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 $K$.

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

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.

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 $b \in L^\infty$, 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 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.

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. 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

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 .

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 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.\

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?