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| = Problems for integro-differential equations with rough coefficients or nonlinear equations = | | ==Registration and content creation== |
| == Hölder estimates for singular integro-differential equations ==
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| Consider an integro-differential equation of the form
| | ===How do I register as a user?=== |
| \[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\]
| | In order to be able to create or edit the pages, you must be a registered user. We welcome all users to register. Unfortunately we have not yet figured out how to set a public registration page and at the same time avoid spammers. So right now the only way to register is by sending an email to one of the administrators: Nestor Guillen, Ray Yang, Russell Schwab or Luis Silvestre. |
| (An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin)
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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.
| | ===How do I edit a page?=== |
| \[ \int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha}, \]
| | There is an edit button on the top that lets you edit each page. You only see it if you are a registered user. |
| 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" />.
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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.
| | ===How do I create a new page?=== |
| | Every time a non existent page is referenced, the link appears red. If you click the red link, you will edit the new page. You need to be a registered user to perform this action. |
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| Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$.
| | ===Where can I learn how to use a wiki?=== |
| | Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software. |
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| == An integral ABP estimate == | | ===How do I write a bibliographical reference?=== |
| | Use the website http://zeteo.info/ to generate the references. |
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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?
| | Look at the pages that are already created as an example of how to make the list of bibliography. |
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| Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
| | ===Are there any rules on what we can write?=== |
| \[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
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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
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| \begin{align*}
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| 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"/>
| | These are the guidelines: |
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| == Holder estimates for parabolic equations with variable order ==
| | # Do not write anything offensive or derogative. |
| | # Avoid using words like ''outstanding'', ''remarkable'', ''groundbreaking'' or ''tour de force'' when describing a result. |
| | # If you think that an article is a triviality or is wrong, it is better not to include it in the citations. |
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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.
| | Making a contribution to the wiki is fairly simple and it can take an arbitrarily small amount of time. Most of the articles are currently not perfect. You can add a paragraph here and there if you have little time. Or you can add a new article which just states a result and hope that someone will pick up the rest. |
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| More precisely, we would like to study a parabolic equation of the form
| | When writing an article, also keep the following priorities in mind. |
| \[ u_t(t,x) = \int_{\R^n} (u(t,x+y) - u(t,x)) K(t,x,y) dy.\]
| | # It has to be '''easy to read'''. This is the top priority. |
| Here $K$ is symmetric (i.e. $K(t,x,y) = K(t,x,-y)$) and satisfies the bounds
| | # It should be clear what is proved and what is not. But see comment below. |
| \[ \frac \lambda {|y|^{n+s(t,x)}} \leq K(t,x,y) \leq \frac \Lambda {|y|^{n+s(t,x)}}.\]
| | # Avoid too much technicalities. If the assumptions of a general result are too complicated, it is ok to just list the major examples. |
| 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?
| | # Give references to the papers where theorems are proved. |
| | # Explain the ideas of the proofs when appropriate. |
| | # If a result is a nonlocal version of a classical theorem, mention it. |
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| == A [[comparison principle]] for $x$-dependent nonlocal equations which are '''not''' in the Levy-Ito form == | | ===Would a user registration ever be revoked?=== |
| 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.
| | Most likely no. Although it could happen if it is used to spam or if one of the guidelines above is grossly disobeyed. |
| \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"/>
| | ===Ok. I registered and want to contribute. What can I do?=== |
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| == Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==
| | Right now, in [[Mwiki:Current events|Current Events]] there is a '''to do''' list. Click on the links and edit the pages. The red links denote that there is a page needed that was not even started. |
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| Consider a [[drift-diffusion equation]] of the form
| | The organization of the wiki is not fully established. We may need some extra index pages or categories. |
| \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\]
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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 $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).
| | There are several pages already. But the wiki is still in a very premature state. Most pages need some more work. The idea of having a wiki is that no version of a page will ever be a final version. However, right now they make that very apparent. |
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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.
| | if you don't know how to start, you can use the pages that are already written as a sample. |
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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 == | | == About the philosophy of the site == |
| 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*}
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| \theta(x,0) &= \theta_0(x) \\
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| \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
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| \end{align*}
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| where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
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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).
| | === Who wrote all this? === |
| | The users of the wiki. Several people. |
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| == Well posedness of the Hilbert flow problem == | | === Whom do the pages belong to? === |
| | Nobody. |
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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.
| | === What if I disagree with something that the wiki says? === |
| \[ \theta_t + \mathrm H\theta \, \theta_x = 0.\]
| | You are free to edit its content. If you do not, we will be offended with you. |
| 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 $s > 1/4$.
| | === Why did you create a wiki? === |
| | If this wiki project works well, it may become a massive reference which is always up to date. It can potentially be better than a book. For that we need several people involved and willing to edit the articles. |
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| = Open problems related to minimal surfaces and free boundaries = | | === Isn't every wiki doomed to fail? === |
| | There are other scientific wiki projects online which can serve as examples: the [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page dispersive wiki], the [http://www.wikiwaves.org/index.php/Main_Page water wave wiki] and also two quantum physics wikis called [http://qwiki.stanford.edu/index.php/Main_Page Qwiki] and [http://www.quantiki.org/wiki/Main_Page Quantiki]. |
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| == Regularity of [[nonlocal minimal surfaces]] ==
| | The success of a wiki page depends on the contributions made by the users. The current experience with scientific wikis shows a questionable level of success. On the other hand, there are non-scientific wikis which are tremendously successful, for example: the [http://wikitravel.org/ travel guide wiki], the [http://harrypotter.wikia.com/wiki/Main_Page Harry Potter wiki], the [http://recipes.wikia.com/wiki/Recipes_Wiki recipes wiki], the [http://starwars.wikia.com/wiki/Main_Page Star Wars wiki], the [http://www.miwiki.net/ Monkey island wiki], the [http://baseball.wikia.com/ Baseball wiki] or the [http://www.mariowiki.com/ Super Mario wiki], among many others. |
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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.
| | It would be interesting to understand this distinction of success between the non-scientific wikis and the scientific ones. |
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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
| | === Why should I spend time writing on this wiki? === |
| # $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
| | If you are a mathematician who has done some research in the area, you definitely want people to know about your results. If you write an easy to read reference in this wiki, that would help more people know about your work and how it is related with other results in the area. Just be careful not to overplay the importance of your own results (or you may be banned from editing again). The appropriate thing to do is to write about all the related results by other people as much as you write about yours. Also remember to follow the rules above in the [[#Are_there_any_rules_on_what_we_can_write|writing guidelines]]. |
| # 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 $n < 8$.
| | If you are a student learning the subject, writing in this wiki may help you understand the topics better (especially if someone comes after you to correct you). Moreover, if you are learning the subject, you probably appreciate the existence of this wiki more than others and are willing to contribute back. |
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| == Optimal regularity for the [[obstacle problem]] for a general integro-differential operator ==
| | There is an interesting video about open science here [http://www.youtube.com/watch?v=DnWocYKqvhw] |
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| Let $u$ be the solution to the [[obstacle problem for the fractional laplacian]],
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| \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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| 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
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| \begin{align*}
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| K(y) &= K(-y) \\
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| \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),
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| \end{align*}
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| 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.
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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 ==
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| {{reflist|refs=
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| <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>
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| <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>
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| <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>
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| <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>
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| <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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| <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>
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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>
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| <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>
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| <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>
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| <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>
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| <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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Registration and content creation
How do I register as a user?
In order to be able to create or edit the pages, you must be a registered user. We welcome all users to register. Unfortunately we have not yet figured out how to set a public registration page and at the same time avoid spammers. So right now the only way to register is by sending an email to one of the administrators: Nestor Guillen, Ray Yang, Russell Schwab or Luis Silvestre.
How do I edit a page?
There is an edit button on the top that lets you edit each page. You only see it if you are a registered user.
How do I create a new page?
Every time a non existent page is referenced, the link appears red. If you click the red link, you will edit the new page. You need to be a registered user to perform this action.
Where can I learn how to use a wiki?
Consult the User's Guide for information on using the wiki software.
How do I write a bibliographical reference?
Use the website http://zeteo.info/ to generate the references.
Look at the pages that are already created as an example of how to make the list of bibliography.
Are there any rules on what we can write?
These are the guidelines:
- Do not write anything offensive or derogative.
- Avoid using words like outstanding, remarkable, groundbreaking or tour de force when describing a result.
- If you think that an article is a triviality or is wrong, it is better not to include it in the citations.
Making a contribution to the wiki is fairly simple and it can take an arbitrarily small amount of time. Most of the articles are currently not perfect. You can add a paragraph here and there if you have little time. Or you can add a new article which just states a result and hope that someone will pick up the rest.
When writing an article, also keep the following priorities in mind.
- It has to be easy to read. This is the top priority.
- It should be clear what is proved and what is not. But see comment below.
- Avoid too much technicalities. If the assumptions of a general result are too complicated, it is ok to just list the major examples.
- Give references to the papers where theorems are proved.
- Explain the ideas of the proofs when appropriate.
- If a result is a nonlocal version of a classical theorem, mention it.
Would a user registration ever be revoked?
Most likely no. Although it could happen if it is used to spam or if one of the guidelines above is grossly disobeyed.
Ok. I registered and want to contribute. What can I do?
Right now, in Current Events there is a to do list. Click on the links and edit the pages. The red links denote that there is a page needed that was not even started.
The organization of the wiki is not fully established. We may need some extra index pages or categories.
There are several pages already. But the wiki is still in a very premature state. Most pages need some more work. The idea of having a wiki is that no version of a page will ever be a final version. However, right now they make that very apparent.
if you don't know how to start, you can use the pages that are already written as a sample.
About the philosophy of the site
Who wrote all this?
The users of the wiki. Several people.
Whom do the pages belong to?
Nobody.
What if I disagree with something that the wiki says?
You are free to edit its content. If you do not, we will be offended with you.
Why did you create a wiki?
If this wiki project works well, it may become a massive reference which is always up to date. It can potentially be better than a book. For that we need several people involved and willing to edit the articles.
Isn't every wiki doomed to fail?
There are other scientific wiki projects online which can serve as examples: the dispersive wiki, the water wave wiki and also two quantum physics wikis called Qwiki and Quantiki.
The success of a wiki page depends on the contributions made by the users. The current experience with scientific wikis shows a questionable level of success. On the other hand, there are non-scientific wikis which are tremendously successful, for example: the travel guide wiki, the Harry Potter wiki, the recipes wiki, the Star Wars wiki, the Monkey island wiki, the Baseball wiki or the Super Mario wiki, among many others.
It would be interesting to understand this distinction of success between the non-scientific wikis and the scientific ones.
Why should I spend time writing on this wiki?
If you are a mathematician who has done some research in the area, you definitely want people to know about your results. If you write an easy to read reference in this wiki, that would help more people know about your work and how it is related with other results in the area. Just be careful not to overplay the importance of your own results (or you may be banned from editing again). The appropriate thing to do is to write about all the related results by other people as much as you write about yours. Also remember to follow the rules above in the writing guidelines.
If you are a student learning the subject, writing in this wiki may help you understand the topics better (especially if someone comes after you to correct you). Moreover, if you are learning the subject, you probably appreciate the existence of this wiki more than others and are willing to contribute back.
There is an interesting video about open science here [1]