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| = Problems for integro-differential equations with rough coefficients or nonlinear equations =
| | A function $f : [0, \infty) \to [0, \infty)$ is said to be an operator monotone function (complete Bernstein function, Nevanlinna-Pick function for the half-line) if $A \ge B \ge 0$ implies $f(A) \ge f(B) \ge 0$ for any self-adjoint matrices $A$, $B$. Many equivalent definitions can be given.<ref name="SSV"/> |
| == Hölder estimates for singular integro-differential equations ==
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|
| Consider an integro-differential equation of the form
| | ==Representation== |
| \[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\] | | A function $f$ is operator monotone if and only if |
| (An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin)
| | \[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \] |
| | for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$. |
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| [[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. | | ==Relation to Bernstein functions== |
| \[ \int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha}, \] | | Operator monotone functions form a subclass of [[Bernstein function]]s. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$: |
| 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" />.
| | \[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \] |
| | has a [[completely monotone function|completely monotone]] density function. In this case |
| | \[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \] |
| | This explains the name complete Bernstein functions. |
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| [[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.
| | ==Holomorphic extension== |
| | | Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that |
| Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$.
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| == An integral ABP estimate == | |
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| 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?
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| Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
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| \[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
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| 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) \\
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| \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).
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| \end{align*}
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| If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?
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| This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.<ref name="GS"/>
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| == Holder estimates for parabolic equations with variable order ==
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| [[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.
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| More precisely, we would like to study a parabolic equation of the form
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| \[ u_t(t,x) = \int_{\R^n} (u(t,x+y) - u(t,x)) K(t,x,y) dy.\]
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| Here $K$ is symmetric (i.e. $K(t,x,y) = K(t,x,-y)$) and satisfies the bounds
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| \[ \frac \lambda {|y|^{n+s(t,x)}} \leq K(t,x,y) \leq \frac \Lambda {|y|^{n+s(t,x)}}.\]
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| 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?
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| == A [[comparison principle]] for $x$-dependent nonlocal equations which are '''not''' in the Levy-Ito form ==
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| Consider two continuous functions $u$ and $v$ such that
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| \begin{align*}
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| u(x) &\leq v(x) \qquad \text{for all $x$ outside some set } \Omega,\\
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| F(x,\{I_\alpha u(x)\}) &\geq F(x,\{I_\alpha v(x)\})\qquad \text{for all $x \in \Omega$}.
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| \end{align*}
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| Is it true that $u \leq v$ in $\Omega$ as well?
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| 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*}
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| I[u] = \int (u(x+z) - u(x) - Du(x)\cdot z 1_{B}(z))\mu_x(dz)
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| \end{align*}
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| where $(\mu_x)_x$ is a family of L\'evy measures, H\"older continous with respect to $x$?
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| Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.<ref name="BI"/>
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| == Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==
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| Consider a [[drift-diffusion equation]] of the form
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| \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]
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| 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 <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 $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).
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| 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.
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| = Open problems for equations related to fluids =
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| == Well posedness of the supercritical [[surface quasi-geostrophic equation]] and related problems ==
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| 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?
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| \begin{align*} | | \begin{align*} |
| \theta(x,0) &= \theta_0(x) \\ | | \Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\ |
| \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty) | | f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\ |
| | \Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 . |
| \end{align*} | | \end{align*} |
| where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
| | Conversely, any function $f$ with above properties is an operator monotone function. |
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| 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).
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| == Well posedness of the Hilbert flow problem ==
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| 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.
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| \[ \theta_t + \mathrm H\theta \, \theta_x = 0.\]
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| 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" />
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| <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
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| \[ \theta_t + \mathrm H\theta \, \theta_x + (-\Delta)^s \theta = 0,\]
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| 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.
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| 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 $s > 1/4$.
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| = Open problems related to minimal surfaces and free boundaries =
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| == Regularity of [[nonlocal minimal surfaces]] ==
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| 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.
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| 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
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| # $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
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| # If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds
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| \[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]
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| When $s$ is sufficiently close to one, such set does not exist if $n < 8$.
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| == Optimal regularity for the [[obstacle problem]] for a general integro-differential operator ==
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| Let $u$ be the solution to the [[obstacle problem for the fractional laplacian]],
| | Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions. |
| \begin{align*}
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| u &\geq \varphi \qquad \text{in } \R^n, \\
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| (-\Delta)^{s/2} u &\geq 0 \qquad \text{in } \R^n, \\
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| (-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\
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| \end{align*}
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| 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
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| \begin{align*}
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| u &\geq \varphi \qquad \text{in } \R^n, \\
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| L u &\leq 0 \qquad \text{in } \R^n, \\
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| L u &= 0 \qquad \text{in } \{u>\varphi\}, \\
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| \end{align*}
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| where $L$ is a [[linear integro-differential operator]], then what is the optimal regularity we can obtain for $u$?
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|
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| 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
| | ==Operator monotone functions of the Laplacian== |
| | Operator monotone functions of the Laplacian are particularly regular examples of translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for an operator monotone $f$ if and only if |
| | \[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + y) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \] |
| | for some $a, b \ge 0$ and $k(z)$ of the form |
| \begin{align*} | | \begin{align*} |
| K(y) &= K(-y) \\
| | k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} \mathrm d t \rho(\mathrm d r) |
| \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*} | | \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.
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| 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.
| | ==References== |
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| UPDATE: This problem has been recently solved by Caffarelli, Ros-Oton, and Serra <ref name="CRS16" />.
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| == Complete understanding of free boundary points in the [[fractional obstacle problem]] ==
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| 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.\
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| # 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.
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| # In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?
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| == References == | |
| {{reflist|refs= | | {{reflist|refs= |
| <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="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</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="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>
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|
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| <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>
| | {{stub}} |
| | |
| <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>
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| <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>
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| <ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
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| first2=Moritz | title=Hölder continuity of harmonic functions with respect to
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| operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
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| doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
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| Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
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| pages=1249–1259}}</ref>
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| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
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| estimates for solutions of integro-differential equations like the fractional
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| Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
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| doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
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| Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
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| pages=1155–1174}}</ref>
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| <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>
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| <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>
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| <ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}}</ref>
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| <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>
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| <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>
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| <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>
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| <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>
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| }} | |