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Purpose
The purpose of this wiki is to stimulate research in elliptic and parabolic nonlocal equations, and to advertise the most interesting open problems in the area.
We would like to write a reference that is easy to read and navigate. It should be convenient to easily find the precise assumptions for which several theorems are proved (for example Harnack inequality or $C^{1,\alpha}$ estimates). It would also be desirable to have some indication of the methods used in the proofs, which may be a non rigorous idea, or an analogy with a classical theorem.
The emphasis of this wiki will be on nonlocal nonlinear (nonbananas?) equations of elliptic and parabolic type. There are several hyperbolic nonlocal equations that are described in the Dispersive wiki (mostly semilinear).
The ultimate goal of this wiki would be to prove for all equations that they are either well posed in the classical sense, or a weak solution and their possible singularities are well understood. The emphasis should then be on regularity results. We may also include some topics only indirectly related to regularity estimates (for example homogenization).
We should give priority in this wiki to the topics that we consider most important. A basic guidance of importance is fully nonlinear $\approx$ quasilinear > semilinear > linear. Unless a linear result has applications to nonlinear equations (for example Holder estimate with measurable coefficients), in which case linear > semilinear. Of course this is only a guideline (Navier-Stokes is semilinear).
We should make an effort to negate the following myths:
- There are no new difficulties in nonlocal equations and everything is proved analogously as in the classical case.
- Nonlocal equations is a field in which one replaces the Laplacian by the fractional Laplacian in whatever equation and writes a paper.
- Nonlocal equations are bizarre and unnatural objects.
- Most equations in nature are local.
- All statements and proofs in nonlocal equations involve gigantic formulas.
what to do?
Eventually we will need to organize all result in categories and also write an introduction to nonlocal equations as a starting point for dummies.
Right now, in Current Events there is a to do list. Click on the links and edit the pages.
if you don't know how to start, you can use the pages that are already written as a sample.
The following is a sample of LeTeX writing.
$
\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
$
We consider, for various values of $s$, the $n$-dimensional integral \begin{align} \label{def:Wns} W_n (s) &:= \int_{[0, 1]^n} \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} \end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.
By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align} \label{eq:W3k} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.