List of results that are fundamentally different to the local case

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\[ \mathrm{div} \, ( A(x) \nabla u) = 0,\]
\[ \mathrm{div} \, ( A(x) \nabla u) = 0,\]
where $\lambda I \leq A(x) \leq \Lambda I$ belong to the space $W^{1,2+\varepsilon}$ for some $\varepsilon > 0$. This is a nontrivial result, since the variational formulation of the problem only gives us a solution in $W^{1,2}$. The result provides an improvement in the integrability of $|\nabla u|$ from $L^2$ to $L^{2+\varepsilon}$.
where $\lambda I \leq A(x) \leq \Lambda I$ belong to the space $W^{1,2+\varepsilon}$ for some $\varepsilon > 0$. This is a nontrivial result, since the variational formulation of the problem only gives us a solution in $W^{1,2}$. The result provides an improvement in the integrability of $|\nabla u|$ from $L^2$ to $L^{2+\varepsilon}$.
 +
 +
The fractional version of the equation consists in a function $u$ so that
 +
\[ \int (u(x)-u(y)) (\eta(x)-\eta(y)) K(x,y) \, dx dy = 0,\]
 +
for all compactly supported, smooth enough, functions $\eta$. It turns out that under the uniform ellipticity assumption $\lambda |x-y|^{-d-2s} \leq K(x,y) \leq \Lambda |x-y|^{-d-2s}$, the solution $u$ turns out to belong to the space $W^{s+\varepsilon,2+\varepsilon}$ <ref name="kuusi2015"/>. The surprising part of the result is that there is an improvement of differentiability. Not only is the power of integrability improved from $2$ to $2+\varepsilon$, but also the order of differentiability is improved from $s$ to $s+\varepsilon$.
== References ==
== References ==

Revision as of 16:33, 6 October 2015

In this page we collect some results in nonlocal equations that contradict the intuition built in analogy with the local case.

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

Contents

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].

Solutions to nonlocal elliptic equations can have interior maximums

Solutions to linear (and nonlinear) integro-differential equations satisfy 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].

In fact, any function $f\in C^k(\overline{B_1})$ can by approximated with a solution to $(-\Delta)^su$ in $B_1$ that vanishes outside a compact set [5]. That is, s-harmonic functions are dense in $C^k_{loc}$. This is clearly in contrast with the rigidity of harmonic functions, and is a purely nonlocal feature.

Boundary regularity of solutions is different from the interior

For second order equations, the boundary regularity of solutions to $\Delta u=0$ is the same as in the interior. For example, a solution to $\Delta u=0$ in $B_1^+$, with $u=0$ in $\{x_n=0\}$, can be extended (by odd reflection) to a solution of $\Delta u=0$ in $B_1$. Thus, in this case the boundary regularity of $u$ just follows from the interior regularity --one has $u\in C^\infty(\overline{B_{1/2}^+})$.

In a general smooth domain $\Omega$ one can flatten the boundary and repeat the previous argument to get that $u\in C^\infty(\overline\Omega)$.

This is not the case for nonlocal equations. Indeed, the function $u(x)=(x_+)^s$ satisfies $(-\Delta)^su=0$ in $(0,\infty)$, and $u=0$ in $(-\infty,0)$. However, $u$ is not even Lipschitz up to the boundary, while all solutions are $C^\infty$ in the interior. This is related to the fact that the odd reflection of $u$ is not anymore a solution to the same equation.

More generally, solutions to $(-\Delta)^su=f$ in $\Omega$, with $u=0$ in $\mathbb R^n\setminus\Omega$, are smooth in the interior of $\Omega$, but not up to the boundary. The optimal Holder regularity is $u\in C^s(\overline\Omega)$. See boundary regularity for integro-differential equations for more details.

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 [6].

Solutions to elliptic linear and translation invariant equations may not be smooth

For second order equations, any solution to an elliptic linear and translation invariant equation $Lu=f$ in $\Omega$ is smooth in the interior whenever $f$ is smooth. For second order equations, $L$ must be of the form $Lu=a_{ij}\partial_{ij}u$, and hence after an affine change of variables it is just the Laplacian $\Delta$.

For nonlocal equations, solutions to $Lu=f$ in $\Omega$, with $f$ smooth, may not be smooth inside $\Omega$, even if $L$ is an elliptic linear and translation invariant operator like \[Lu(x)=\int_{\mathbb R^n}\bigl(u(x+y)-u(x)\bigr)K(y)dy,\] with $K(y)=K(-y)$ and satisfying \[\frac{\lambda}{|y|^{n+2s}}\leq K(y)\leq \frac{\Lambda}{|y|^{n+2s}}.\] It was proved in [7] that there exist a solution to $Lu=0$ in $B_1$, with $u\in L^\infty(\R^n)$, which is not $C^{2s+\epsilon}(B_{1/2})$ for any $\epsilon>0$. The counterexample can be constructed even in dimension 1, and it is very related to the regularity of the kernel $K$.

Related to this, it was shown in [8] that there is a $C^\infty$ domain $\Omega$ and an operator of the form \[Lu(x)=\int_{\mathbb R^n}\bigl(u(x+y)-u(x)\bigr)\frac{a(y/|y|)}{|y|^{n+2s}}\,dy,\] for which the solution to $Lu=1$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, is not $C^{3s+\epsilon}$ inside $\Omega$ for any $\epsilon>0$. See the survey [9] for more details.


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 [10].

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 [11].

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 [12].

Improved differentiability of solutions to integro-differential equations in divergence form

A classical theorem asserts that solution to uniformly elliptic equations in divergence form \[ \mathrm{div} \, ( A(x) \nabla u) = 0,\] where $\lambda I \leq A(x) \leq \Lambda I$ belong to the space $W^{1,2+\varepsilon}$ for some $\varepsilon > 0$. This is a nontrivial result, since the variational formulation of the problem only gives us a solution in $W^{1,2}$. The result provides an improvement in the integrability of $|\nabla u|$ from $L^2$ to $L^{2+\varepsilon}$.

The fractional version of the equation consists in a function $u$ so that \[ \int (u(x)-u(y)) (\eta(x)-\eta(y)) K(x,y) \, dx dy = 0,\] for all compactly supported, smooth enough, functions $\eta$. It turns out that under the uniform ellipticity assumption $\lambda |x-y|^{-d-2s} \leq K(x,y) \leq \Lambda |x-y|^{-d-2s}$, the solution $u$ turns out to belong to the space $W^{s+\varepsilon,2+\varepsilon}$ [13]. The surprising part of the result is that there is an improvement of differentiability. Not only is the power of integrability improved from $2$ to $2+\varepsilon$, but also the order of differentiability is improved from $s$ to $s+\varepsilon$.

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. Dipierro, Serena; Savin, Ovidiu; Valdinoci, Enrico, "All functions are locally $s$-harmonic up to a small error", arXiv preprint arXiv:1404.3652 
  6. Rang, Marcus; Kassmann, Moritz; Schwab, Russell W, "H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence", arXiv preprint arXiv:1306.0082 
  7. Serra, Joaquim, "$C^{2s+\alpha}$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels", arXiv preprint 
  8. Ros-Oton, Xavier; Valdinoci, Enrico, "The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains", arXiv preprint 
  9. Ros-Oton, Xavier, "Nonlocal elliptic equations in bounded domains: a survey", arXiv preprint 
  10. 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 
  11. Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X 
  12. 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 
  13. Kuusi, Tuomo; Mingione, Giuseppe; Sire, Yannick (2015), "Nonlocal self-improving properties", Anal. PDE 8: 57--114, doi:10.2140/apde.2015.8.57, ISSN 2157-5045, http://dx.doi.org/10.2140/apde.2015.8.57 
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