List of results that are fundamentally different to the local case: Difference between revisions

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The intuitive explanation is that $(-\Delta)^s u$ satisfies an extra elliptic equation in terms of its Laplacian to the power $1-s$, and that equation provides the extra regularity <ref name="S"/>.
The intuitive explanation is that $(-\Delta)^s u$ satisfies an extra elliptic equation in terms of its Laplacian to the power $1-s$, and that equation provides the extra regularity <ref name="S"/>.
=== Nonlocal elliptic equations can have interior maximums ===
A solution to a [[fully nonlinear integro-differential equation]] satisfies a ''nonlocal'' maximum principle: they cannot have a ''global'' maximum or minumum in the interior of the domain of the equation. Local extrema are possible.
This is related to the fact that Dirichlet boundary conditions have to be given in the whole complement of the Domain and not only on its boundary. It is also related to the failure in general of the classical [[Harnack inequality]] unless the possitivity of the function is assumed in the full space <ref name="K"/>.


== References ==
== References ==
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<ref name="CR">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire | url=http://dx.doi.org/10.1016/j.crma.2009.10.012 | doi=10.1016/j.crma.2009.10.012 | year=2009 | journal=Comptes Rendus Mathématique. Académie des Sciences. Paris | issn=1631-073X | volume=347 | issue=23 | pages=1361–1366}}</ref>
<ref name="CR">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire | url=http://dx.doi.org/10.1016/j.crma.2009.10.012 | doi=10.1016/j.crma.2009.10.012 | year=2009 | journal=Comptes Rendus Mathématique. Académie des Sciences. Paris | issn=1631-073X | volume=347 | issue=23 | pages=1361–1366}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=The classical Harnack inequality fails for non-local operators | year=Preprint}}</ref>
}}
}}

Revision as of 18:49, 15 July 2011

In this page we collect some results in nonlocal equations when things behave very differently compared to the local counterpart. A result makes it to this list if it is somewhat suprising or counterintuitive.

The list is very incomplete right now. Please help expand it by editing it.

Traveling fronts in Fisher-KPP equations with fractional diffusion

Let us consider the reaction diffusion equation \[ u_t + (-\Delta)^s u = f(u), \] with a Fisher-KPP type of nonlinearity (for example $f(u) = u(1-u)$). In the local diffusion case, the stable state $u=1$ invades the unstable state $u=0$ at a constant speed. In the nonlocal case (any $s<1$), the invasion holds at an exponential rate.

The explanation of the difference can be understood intuitively from the fact that the fat tails in the fractional heat kernels make diffusion happen at a much faster rate [1].

Optimal regularity for the fractional obstacle problem

Given a function $\varphi$, the obstacle problem consists in the solution to an equation of the form \[ \min((-\Delta)^s u , u-\varphi) = 0.\]

If $\varphi$ is smooth enough, the solution $u$ to the obstacle problem will be $C^{1,s}$ and no better. There is a big difference between the case $s=1$ and $s<1$ which makes the proof fundamentally different. In the classical case $s=1$, the optimal regularity matches the scaling of the equation. The classical proof of optimal regularity is to show an upper bound in the separation of $u$ from the obstacle in the unit ball and then just scale it. In the fractional case $s<1$, this method only gives $C^{2s}$ regularity, which matches the scaling of the equation. It is somewhat surprising that a better regularity result holds and it requires a different method for the proof.

The intuitive explanation is that $(-\Delta)^s u$ satisfies an extra elliptic equation in terms of its Laplacian to the power $1-s$, and that equation provides the extra regularity [2].

Nonlocal elliptic equations can have interior maximums

A solution to a fully nonlinear integro-differential equation satisfies a nonlocal maximum principle: they cannot have a global maximum or minumum in the interior of the domain of the equation. Local extrema are possible.

This is related to the fact that Dirichlet boundary conditions have to be given in the whole complement of the Domain and not only on its boundary. It is also related to the failure in general of the classical Harnack inequality unless the possitivity of the function is assumed in the full space [3].

References

  1. Cabré, Xavier; Roquejoffre, Jean-Michel (2009), "Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire", Comptes Rendus Mathématique. Académie des Sciences. Paris 347 (23): 1361–1366, doi:10.1016/j.crma.2009.10.012, ISSN 1631-073X, http://dx.doi.org/10.1016/j.crma.2009.10.012 
  2. Silvestre, Luis (2007), "Regularity of the obstacle problem for a fractional power of the Laplace operator", Communications on Pure and Applied Mathematics 60 (1): 67–112, doi:10.1002/cpa.20153, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20153 
  3. Kassmann, Moritz (Preprint), The classical Harnack inequality fails for non-local operators