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

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This procedure can also be carried out to approximate viscosity solutions to fully nonlinear elliptic equations with smooth solutions of an approximate equation. However, the approximated equation is integro-differential <ref name="CS4"/>.
This procedure can also be carried out to approximate viscosity solutions to fully nonlinear elliptic equations with smooth solutions of an approximate equation. However, the approximated equation is integro-differential <ref name="CS4"/>.
=== For some equations, the weak Harnack inequality may hold while the full [[Harnack inequality]] does not ===
The weak Harnack inequality relates the minimum of a positive supersolution to an elliptic equation to its $L^p$ norm. It is an important step used to derive [[Hölder estimates]] and also the usual [[Harnack inequality]]. However, there are examples of non local elliptic equation for which the weak Harnack inequality and Hölder estimates hold, whereas the classical Harnack inequality does not. There is a discussion about this fact in an article by Moritz Kassmann, Marcus Rang and Russell Schwab <ref name="rang2013h" />.


== References ==
== References ==
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<ref name="CS3">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=The Evans-Krylov theorem for non local fully non linear equations | year=to appear | journal=[[Annals of Mathematics]] | issn=0003-486X}}</ref>
<ref name="CS3">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=The Evans-Krylov theorem for non local fully non linear equations | year=to appear | journal=[[Annals of Mathematics]] | issn=0003-486X}}</ref>
<ref name="CS4">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Nonlinear partial differential equations and related topics | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Amer. Math. Soc. Transl. Ser. 2 | year=2010 | volume=229 | chapter=Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations | pages=67–85}}</ref>
<ref name="CS4">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Nonlinear partial differential equations and related topics | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Amer. Math. Soc. Transl. Ser. 2 | year=2010 | volume=229 | chapter=Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations | pages=67–85}}</ref>
<ref name="rang2013h">{{Citation | last1=Rang | first1= Marcus | last2=Kassmann | first2= Moritz | last3=Schwab | first3= Russell W | title=H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence | journal=arXiv preprint arXiv:1306.0082}}</ref>
}}
}}

Revision as of 11:12, 16 July 2014

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 have exponential speed

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][2].

The optimal regularity for the fractional obstacle problem exceeds the scaling of the equation

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 [3].

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 positivity of the function is assumed in the full space [4].

Viscosity solutions can be evaluated at points

The concept of viscosity solutions is developed in order to make sense of an elliptic equation even for continuous functions for which the equation cannot be evaluated classically at points. The idea is to use test functions whose graphs are tangent to the graphs of the weak solution at some point, and then evaluate the equation on that test function. The ellipticity property tells us that the value of the equation for that test function at the point of contact must have certain sign, and this is the condition that a viscosity solution fulfill.

It turns out that for a large class of fully nonlinear integro-differential equations, every time a viscosity solution can be touched by a smooth test function at a point, then the equation can be evaluated classically for the original function at that point [5].

Viscosity solutions to fully nonlinear integro-differential equations can be approximated with classical solutions

It is a very classical trick that if we have a weak solution to a linear PDE with constant coefficients, we can approximate it with a smooth solution via a simple mollification. For nonlinear equations this trick is no longer available and we are always forced to deal with the technical difficulties of viscosity solutions. This is an apparent difficulty for example when proving regularity estimates, since in general we cannot derive them an a priori estimate for a classical solution. On the other hand, viscosity solutions to fully nonlinear integro-differential equations can be approximated by $C^2$ solutions to approximate equations [6].

This procedure can also be carried out to approximate viscosity solutions to fully nonlinear elliptic equations with smooth solutions of an approximate equation. However, the approximated equation is integro-differential [7].

For some equations, the weak Harnack inequality may hold while the full Harnack inequality does not

The weak Harnack inequality relates the minimum of a positive supersolution to an elliptic equation to its $L^p$ norm. It is an important step used to derive Hölder estimates and also the usual Harnack inequality. However, there are examples of non local elliptic equation for which the weak Harnack inequality and Hölder estimates hold, whereas the classical Harnack inequality does not. There is a discussion about this fact in an article by Moritz Kassmann, Marcus Rang and Russell Schwab [8].

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. Cabré, Xavier; Roquejoffre, Jean-Michel (to appear), "The influence of fractional diffusion in Fisher-KPP equations", Comm. Math. Phys., http://arxiv.org/abs/1202.6072 
  3. 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 
  4. Kassmann, Moritz (Preprint), The classical Harnack inequality fails for non-local operators 
  5. Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 
  6. Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X 
  7. Caffarelli, Luis; Silvestre, Luis (2010), "Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations", Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, 229, Providence, R.I.: American Mathematical Society, pp. 67–85 
  8. Rang, Marcus; Kassmann, Moritz; Schwab, Russell W, "H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence", arXiv preprint arXiv:1306.0082