To Do List and Hölder estimates: Difference between pages

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== Things that need to be done ==
Hölder continuity of the solutions can sometimes be proved only from ellipticity
assumptions on the equation, without depending on smoothness of the
coefficients. This allows great flexibility in terms of applications of the
result. The corresponding result for elliptic equations of second order is the
[[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser theorem]] in the divergence form.


We need to come up with some organization for the articles.
The Hölder estimates are closely related to the [[Harnack inequality]]. In most cases, one can deduce the Hölder estimates from the Harnack inequality. However, there are simple example of integro-differential equations for which the Hölder estimates hold and the Harnack inequality does not <ref name="rang2013h" /> <ref name="bogdan2005harnack" />.


The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki.
There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
and [[Krylov-Safonov theorem]]. The former uses variational techniques and is
stated in terms of Dirichlet forms. The latter is based on comparison
principles.


* We better start thinking hard about writing the [[Introduction to nonlocal equations]]. We gotta start somewhere, so any random idea or small thing you want to write should go in there. This is high priority, since if someone reads one page of this wiki, it will likely be this one.
A Hölder estimate says that a solution to an integro-differential equation with rough coefficients
$L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
(small). It is very important when an estimate allows for a very rough dependence of
$L_x$ with respect to $x$, since the result then applies to the linearization of
(fully) nonlinear equations without any extra a priori estimate. On the other
hand, the linearization of a [[fully nonlinear integro-differential equation]] (for example the [[Isaacs equation]] or the [[Bellman equation]]) would inherit the initial assumptions regarding for the kernels with
respect to $y$. Therefore, smoothness (or even structural) assumptions for the
kernels with respect to $y$ can be made keeping such result applicable.


* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]].
In the non variational setting the integro-differential operators $L_x$ are
assumed to belong to some family, but no continuity is assumed for its
dependence with respect to $x$. Typically, $L_x u(x)$ has the form
$$ L_x u(x) = a_{ij}(x) \partial_{ij} u + b(x) \cdot \nabla u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
K(x,y) \, dy$$
Within the context of nonlocal equations, we would be interested on a regularization effect caused by the integral term and not the second order part of the equation. Because of that, the coefficients $a_{ij}(x)$ are usually assumed to be zero.


* Fractional curvatures in conformal geometry.  
Since [[linear integro-differential operators]] allow for a great flexibility of
equations, there are several variations on the result: different assumptions on
the kernels, mixed local terms, evolution equations, etc. The linear equation
with rough coefficients is equivalent to the function $u$ satisfying two
inequalities for the [[extremal operators]] corresponding to the family of
operators $L$, which stresses the nonlinear character of the estimates.


* We need  to explain further the [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geometric Scattering Theory]], [[Ambient Metric Construction]], [[GJMS Operators]] and [[Singular Yamabe Problem]]..)
As with other estimates in this field too, some Hölder estimates blow up as the
order of the equation converges to two, and others pass to the limit. The
blow-up is a matter of the techniques used in the proof. Only estimates which
are robust are a true generalization of either the [[De Giorgi-Nash-Moser theorem]] or
[[Krylov-Safonov theorem]].  


* It would be wise (once the wiki is more mature) to add pages about the [[Boltzmann equation]], since it is one of the more "classical" and better known integro-differential equations.
== The general statement ==


* Pages about [[Homogenization]] (local and nonlocal) should appear here too.
=== Elliptic form ===
The general form of the Hölder estimates for an elliptic problem say that if we have an equation which holds in a domain, and the solution is globally bounded, then the solution is Hölder continuous in the interior of the domain. Typically this is stated in the following form: if $u : \R^d \to \R$ solves
\[
L(u,x) = 0 \ \ \text{in } B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]


* Given the recent works of Osher/Gilboa and Bertozzi/Flenner on Ginzburg-Landau  on graphs we should have an article on the natural similarities between non-local operators and [[elliptic operators on graphs]].
There is no lack of generality in assuming that $L$ is a '''linear''' integro-differential operator, provided that there is no regularity assumption on its $x$ dependence.


* [[nonlocal image processing]].
For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
\begin{align*}
M^+u \geq 0 \ \ \text{in } B_1, \\
M^-u \leq 0 \ \ \text{in } B_1.
\end{align*}
where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.


* [[Aggregation equation]].
=== Parabolic form ===
The general form of the Hölder estimates for a parabolic problem is also an interior regularity statement for solutions of a parabolic equation. Typically this is stated in the following form: if $u : \R^d \times (-1,0] \to \R$ solves
\[
u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
\]
and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
\[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]


* [[Keller-Segel equation]].
== List of results ==


== (partially) Completed tasks ==
There are several Hölder estimates for elliptic and parabolic integro-differential equations which have been obtained. Here we list some of these results with a brief description of their main assumptions.


* A sort of [[Starting page]] that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.)
=== Variational equations ===
A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
$E(u,v)$ satisfying
$$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx
\, dy $$
on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note
that $K$ can be assumed to be symmetric because the skew-symmetric part
of $K$ would be ignored by the bilinear form.  


* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]].
Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler
equation
$$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
which should be understood in the sense of distributions.


* Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Alexadroff-Bakelman-Pucci estimates]], some reference to [[free boundary problems]].
It is known that the gradient flow of a Dirichlet form (parabolic version of the
result) becomes instantaneously Hölder continuous <ref name="CCV"/>. The method
of the proof builds an integro-differential version of the parabolic De Giorgi
technique that was developed for the study of critical [[Surface quasi-geostrophic equation]].  


* A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]].
At some point in the original proof of De Giorgi, it is used that the
characteristic functions of a set of positive measure do not belong to $H^1$.
Moreover, a quantitative estimate is required about the measure of
''intermediate'' level sets for $H^1$ functions. In the integro-differential
context, the required statement to carry out the proof would be the same with
the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would
even require a non trivial proof for $s$ close to $2$. The difficulty is
bypassed though an argument that takes advantage of the nonlocal character of
the equation, and hence the estimate blows up as the order approaches two.


* Some discussion on models involving [[Levy processes]] and [[stochastic control]].
In the stationary case, it is known that minimizers of Dirichlet forms are
Hölder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser theorem]] to the
nonlocal setting <ref name="K"/>. In this result, the constants do not blow up as the order of the equation goes to two.


* Some references to equations from fluids including the [[surface quasi-geostrophic equation]].
{{stub}}


* A page about [[semilinear equations]] including the [[surface quasi-geostrophic equation]] and also some form of KPP.
=== Non variational equations ===
The results below are nonlocal versions of [Krylov-Safonov theorem].  No regularity needs to be
assumed for $K$ with respect to $x$.


* [[Nonlocal porous medium equation]].
* The first result was obtained by Bass and Levin using probabilistic techniques.<ref
name="BL"/> It applies to elliptic integro-differential equations with symmetric and uniformly elliptic kernels (bounded pointwise). The constants obtained in the estimates are not uniform as the order of the equation goes to two. An extension of this result was obtained by Song and Vondracev <ref name="song2004" />.


* [[drift-diffusion equations]].
The estimate says the following. Assume a bounded function $u: \R^n \to \R$ solves
$$ \int_{\R^n} (u(x+y) - u(x)) K(x,y) \mathrm{d}y = 0 \qquad \text{for all $x$ in } B_1, $$
where $K$ satisfies the symmetry assumption $K(x,y) = K(x,-y)$ and
\begin{equation} \label{pointwisebound}
\frac{\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s}} \qquad \text{for all } x,y \in \R^n,
\end{equation}
where $s \in (0,2)$ and $\Lambda \geq \lambda > 0$ are given parameters. Then
\[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^n)}.\]


* [[Nonlocal minimal surfaces]] and [[Nonlocal mean curvature flow]].
* A result by Bass and Kassmann also uses probabilistic techniques.<ref
name="BL"/> It applies to elliptic integro-differential equations with a rather general set of assumptions in the kernels. The main novelty is that the order of the equation may vary (continuously) from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
* The first purely analytic proof was obtained by Luis Silvestre <ref
name="S"/>. The assumptions on the kernel are similar to those of Bass and Kassmann except that the order of the equation may change abruptly from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
* The first estimate which remains uniform as the order of the equation goes to two was obtained by Caffarelli and Silvestre <ref name="CS"/>. The equations here are elliptic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}).
* An estimate for parabolic equations of order one with a bounded drift was obtained by Silvestre <ref name="silvestre2011differentiability"/>. The equation here is parabolic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}). The order of the equation is set to be one, but the proof also gives the estimate for any order greater than one, or also less than one if there is no drift. The constants in the estimates blow up as the order of the equation converges to two.
* Davila and Chang-Lara studied equations with nonsymmetric kernels, elliptic and parabolic. Their first result is for elliptic equations so that the odd part of the kernels is of lower order compared to their symmetric part <ref name="lara2012regularity" />. A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.<ref name="lara2014regularity" />. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels <ref name="chang2014h" />. For these three results,the kernels are required to be uniformly elliptic (pointwise bounded as in \ref{pointwisebound}).
* The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli <ref name="bjorland2012" /> for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab <ref name="rang2013h"/>.
* A generalization of all previous results was obtained by Schwab and Silvestre <ref name="schwab2014regularity" />. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described [[Linear_integro-differential_operator#More singular/irregular kernels|here]].
* A non scale invariant family of kernels was studied by Kassmann and Mimica <ref name="kassmann2013intrinsic"/>. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $|y|$ which is not necessarily power-like.


* There is also plenty of work on [[Dislocation dynamics]] that we ought to add later on.
== Other variants ==


* Phase transitions involving non-local interactions, in particular, pages about [[Particle Systems]], discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.
* There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
* If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a very general class of integral equations <ref name="barles2011" />.


* It is convenient to have a [[mini second order elliptic wiki]] inside this wiki.


* [[open problems]].
== References ==
 
{{reflist|refs=
* Having a [[list of equations]] may make it easier to navigate the wiki.
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
 
estimates for solutions of integro-differential equations like the fractional
== Other ideas ==
Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
 
doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
 
Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
* Fill up the list of [[upcoming events]] such as conferences, workshops, summer schools.
pages=1155–1174}}</ref>
 
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre |
* [[User:Nestor|Nestor]] has started a [[Literature on Nonlocal Equations]] to dump there all papers we want to reference or are already referencing on the wiki.
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>
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
first2=Chi Hin | last3=Vasseur | first3=Alexis | title= |
doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the
American Mathematical Society]] | issn=0894-0347 | issue=24 |
pages=849–869}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori
estimates for integro-differential operators with measurable kernels |
url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6
| year=2009 | journal=Calculus of Variations and Partial Differential Equations
| issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
first2=Moritz | title=Hölder continuity of harmonic functions with respect to
operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
pages=1249–1259}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin |
first2=David A. | title=Harnack inequalities for jump processes |
url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 |
year=2002 | journal=Potential Analysis. An International Journal Devoted to the
Interactions between Potential Theory, Probability Theory, Geometry and
Functional Analysis | issn=0926-2601 | volume=17 | issue=4 |
pages=375–388}}</ref>
<ref name="lara2011regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | journal=Calculus of Variations and Partial Differential Equations | year=2011 | pages=1--34}}</ref>
<ref name="zacher2013">{{Citation | last1=Zacher | first1= Rico | title=A De Giorgi--Nash type theorem for time fractional diffusion equations | url=http://dx.doi.org/10.1007/s00208-012-0834-9 | journal=Math. Ann. | issn=0025-5831 | year=2013 | volume=356 | pages=99--146 | doi=10.1007/s00208-012-0834-9}}</ref>
<ref name="barles2011">{{Citation | last1=Barles | first1= Guy | last2=Chasseigne | first2= Emmanuel | last3=Imbert | first3= Cyril | title=H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations | url=http://dx.doi.org/10.4171/JEMS/242 | journal=J. Eur. Math. Soc. (JEMS) | issn=1435-9855 | year=2011 | volume=13 | pages=1--26 | doi=10.4171/JEMS/242}}</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>
<ref name="bogdan2005harnack">{{Citation | last1=Bogdan | first1= Krzysztof | last2=Sztonyk | first2= Pawe\l | title=Harnack’s inequality for stable Lévy processes | journal=Potential Analysis | year=2005 | volume=22 | pages=133--150}}</ref>
<ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref>
<ref name="kassmann2013intrinsic">{{Citation | last1=Kassmann | first1= Moritz | last2=Mimica | first2= Ante | title=Intrinsic scaling properties for nonlocal operators | journal=arXiv preprint arXiv:1310.5371}}</ref>
<ref name="song2004">{{Citation | last1=Song | first1= Renming | last2=Vondra\vcek | first2= Zoran | title=Harnack inequality for some classes of Markov processes | url=http://dx.doi.org/10.1007/s00209-003-0594-z | journal=Math. Z. | issn=0025-5874 | year=2004 | volume=246 | pages=177--202 | doi=10.1007/s00209-003-0594-z}}</ref>
<ref name="silvestre2011differentiability">{{Citation | last1=Silvestre | first1= Luis | title=On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion | journal=Advances in mathematics | year=2011 | volume=226 | pages=2020--2039}}</ref>
<ref name="lara2012regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of nonlocal, nonsymmetric equations | year=2012 | volume=29 | pages=833--859}}</ref>
<ref name="chang2014h">{{Citation | last1=Chang-Lara | first1= Hector | last2=Davila | first2= Gonzalo | title=H$\backslash$" older estimates for non-local parabolic equations with critical drift | journal=arXiv preprint arXiv:1408.0676}}</ref>
<ref name="lara2014regularity">{{Citation | last1=Lara | first1= Héctor Chang | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | journal=Calculus of Variations and Partial Differential Equations | year=2014 | volume=49 | pages=139--172}}</ref>
<ref name="bjorland2012">{{Citation | last1=Bjorland | first1= C. | last2=Caffarelli | first2= L. | last3=Figalli | first3= A. | title=Non-local gradient dependent operators | url=http://dx.doi.org/10.1016/j.aim.2012.03.032 | journal=Adv. Math. | issn=0001-8708 | year=2012 | volume=230 | pages=1859--1894 | doi=10.1016/j.aim.2012.03.032}}</ref>
}}

Revision as of 16:56, 8 April 2015

Hölder continuity of the solutions can sometimes be proved only from ellipticity assumptions on the equation, without depending on smoothness of the coefficients. This allows great flexibility in terms of applications of the result. The corresponding result for elliptic equations of second order is the Krylov-Safonov theorem in the non-divergence form, or the De Giorgi-Nash-Moser theorem in the divergence form.

The Hölder estimates are closely related to the Harnack inequality. In most cases, one can deduce the Hölder estimates from the Harnack inequality. However, there are simple example of integro-differential equations for which the Hölder estimates hold and the Harnack inequality does not [1] [2].

There are integro-differential versions of both De Giorgi-Nash-Moser theorem and Krylov-Safonov theorem. The former uses variational techniques and is stated in terms of Dirichlet forms. The latter is based on comparison principles.

A Hölder estimate says that a solution to an integro-differential equation with rough coefficients $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$ (small). It is very important when an estimate allows for a very rough dependence of $L_x$ with respect to $x$, since the result then applies to the linearization of (fully) nonlinear equations without any extra a priori estimate. On the other hand, the linearization of a fully nonlinear integro-differential equation (for example the Isaacs equation or the Bellman equation) would inherit the initial assumptions regarding for the kernels with respect to $y$. Therefore, smoothness (or even structural) assumptions for the kernels with respect to $y$ can be made keeping such result applicable.

In the non variational setting the integro-differential operators $L_x$ are assumed to belong to some family, but no continuity is assumed for its dependence with respect to $x$. Typically, $L_x u(x)$ has the form $$ L_x u(x) = a_{ij}(x) \partial_{ij} u + b(x) \cdot \nabla u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy$$ Within the context of nonlocal equations, we would be interested on a regularization effect caused by the integral term and not the second order part of the equation. Because of that, the coefficients $a_{ij}(x)$ are usually assumed to be zero.

Since linear integro-differential operators allow for a great flexibility of equations, there are several variations on the result: different assumptions on the kernels, mixed local terms, evolution equations, etc. The linear equation with rough coefficients is equivalent to the function $u$ satisfying two inequalities for the extremal operators corresponding to the family of operators $L$, which stresses the nonlinear character of the estimates.

As with other estimates in this field too, some Hölder estimates blow up as the order of the equation converges to two, and others pass to the limit. The blow-up is a matter of the techniques used in the proof. Only estimates which are robust are a true generalization of either the De Giorgi-Nash-Moser theorem or Krylov-Safonov theorem.

The general statement

Elliptic form

The general form of the Hölder estimates for an elliptic problem say that if we have an equation which holds in a domain, and the solution is globally bounded, then the solution is Hölder continuous in the interior of the domain. Typically this is stated in the following form: if $u : \R^d \to \R$ solves \[ L(u,x) = 0 \ \ \text{in } B_1, \] and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$, \[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]

There is no lack of generality in assuming that $L$ is a linear integro-differential operator, provided that there is no regularity assumption on its $x$ dependence.

For non variational problems, in order to adapt the situation to the viscosity solution framework, the equation may be replaced by two inequalities. \begin{align*} M^+u \geq 0 \ \ \text{in } B_1, \\ M^-u \leq 0 \ \ \text{in } B_1. \end{align*} where $M^+$ and $M^-$ are extremal operators with respect to some class.

Parabolic form

The general form of the Hölder estimates for a parabolic problem is also an interior regularity statement for solutions of a parabolic equation. Typically this is stated in the following form: if $u : \R^d \times (-1,0] \to \R$ solves \[ u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1, \] and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$, \[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]

List of results

There are several Hölder estimates for elliptic and parabolic integro-differential equations which have been obtained. Here we list some of these results with a brief description of their main assumptions.

Variational equations

A typical example of a symmetric nonlocal Dirichlet form is a bilinear form $E(u,v)$ satisfying $$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx \, dy $$ on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note that $K$ can be assumed to be symmetric because the skew-symmetric part of $K$ would be ignored by the bilinear form.

Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler equation $$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$ which should be understood in the sense of distributions.

It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Hölder continuous [3]. The method of the proof builds an integro-differential version of the parabolic De Giorgi technique that was developed for the study of critical Surface quasi-geostrophic equation.

At some point in the original proof of De Giorgi, it is used that the characteristic functions of a set of positive measure do not belong to $H^1$. Moreover, a quantitative estimate is required about the measure of intermediate level sets for $H^1$ functions. In the integro-differential context, the required statement to carry out the proof would be the same with the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would even require a non trivial proof for $s$ close to $2$. The difficulty is bypassed though an argument that takes advantage of the nonlocal character of the equation, and hence the estimate blows up as the order approaches two.

In the stationary case, it is known that minimizers of Dirichlet forms are Hölder continuous by adapting Moser's proof of De Giorgi-Nash-Moser theorem to the nonlocal setting [4]. In this result, the constants do not blow up as the order of the equation goes to two.

This article is a stub. You can help this nonlocal wiki by expanding it.

Non variational equations

The results below are nonlocal versions of [Krylov-Safonov theorem]. No regularity needs to be assumed for $K$ with respect to $x$.

  • The first result was obtained by Bass and Levin using probabilistic techniques.[5] It applies to elliptic integro-differential equations with symmetric and uniformly elliptic kernels (bounded pointwise). The constants obtained in the estimates are not uniform as the order of the equation goes to two. An extension of this result was obtained by Song and Vondracev [6].

The estimate says the following. Assume a bounded function $u: \R^n \to \R$ solves $$ \int_{\R^n} (u(x+y) - u(x)) K(x,y) \mathrm{d}y = 0 \qquad \text{for all $x$ in } B_1, $$ where $K$ satisfies the symmetry assumption $K(x,y) = K(x,-y)$ and \begin{equation} \label{pointwisebound} \frac{\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s}} \qquad \text{for all } x,y \in \R^n, \end{equation} where $s \in (0,2)$ and $\Lambda \geq \lambda > 0$ are given parameters. Then \[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^n)}.\]

  • A result by Bass and Kassmann also uses probabilistic techniques.[5] It applies to elliptic integro-differential equations with a rather general set of assumptions in the kernels. The main novelty is that the order of the equation may vary (continuously) from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
  • The first purely analytic proof was obtained by Luis Silvestre [7]. The assumptions on the kernel are similar to those of Bass and Kassmann except that the order of the equation may change abruptly from point to point. The constants obtained in the estimates are not uniform as the order of the equation goes to two.
  • The first estimate which remains uniform as the order of the equation goes to two was obtained by Caffarelli and Silvestre [8]. The equations here are elliptic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}).
  • An estimate for parabolic equations of order one with a bounded drift was obtained by Silvestre [9]. The equation here is parabolic, with symmetric and uniformly elliptic kernels (pointwise bounded as in \ref{pointwisebound}). The order of the equation is set to be one, but the proof also gives the estimate for any order greater than one, or also less than one if there is no drift. The constants in the estimates blow up as the order of the equation converges to two.
  • Davila and Chang-Lara studied equations with nonsymmetric kernels, elliptic and parabolic. Their first result is for elliptic equations so that the odd part of the kernels is of lower order compared to their symmetric part [10]. A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.[11]. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels [12]. For these three results,the kernels are required to be uniformly elliptic (pointwise bounded as in \ref{pointwisebound}).
  • The first relaxation of the pointwise bound \eqref{pointwisebound} on the kernels was given by Bjorland, Caffarelli and Figalli [13] for elliptic equations. They consider symmetric kernels which are bounded above everywhere, but are bounded below only in a cone of directions. A similar result is obtained by Kassmann, Rang and Schwab [1].
  • A generalization of all previous results was obtained by Schwab and Silvestre [14]. They consider parabolic equations. The estimates stay uniform as the order of the equation goes to two. The kernels are assumed to be bounded above only in average and bounded below only in sets of positive density. There is no symmetry assumption. It corresponds to the class of operators described here.
  • A non scale invariant family of kernels was studied by Kassmann and Mimica [15]. They study elliptic equations with symmetric kernels that satisfy pointwise bounds. However, the assumption is different from \eqref{pointwisebound} in the sense that $K(x,y)$ is required to be comparable to a function of $|y|$ which is not necessarily power-like.

Other variants

  • There are Holder estimates for equations in divergence form that are non local in time [16]
  • If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a very general class of integral equations [17].


References

  1. 1.0 1.1 Rang, Marcus; Kassmann, Moritz; Schwab, Russell W, "H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence", arXiv preprint arXiv:1306.0082 
  2. Bogdan, Krzysztof; Sztonyk, Pawe\l (2005), "Harnack’s inequality for stable Lévy processes", Potential Analysis 22: 133--150 
  3. Caffarelli, Luis; Chan, Chi Hin; Vasseur, Alexis (2011), [[Journal of the American Mathematical Society]] (24): 849–869, doi:10.1090/S0894-0347-2011-00698-X, ISSN 0894-0347 
  4. Kassmann, Moritz (2009), [http://dx.doi.org/10.1007/s00526-008-0173-6 "A priori estimates for integro-differential operators with measurable kernels"], Calculus of Variations and Partial Differential Equations 34 (1): 1–21, doi:10.1007/s00526-008-0173-6, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-008-0173-6 
  5. 5.0 5.1 Bass, Richard F.; Levin, David A. (2002), "Harnack inequalities for jump processes", Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis 17 (4): 375–388, doi:10.1023/A:1016378210944, ISSN 0926-2601, http://dx.doi.org/10.1023/A:1016378210944 
  6. Song, Renming; Vondra\vcek, Zoran (2004), "Harnack inequality for some classes of Markov processes", Math. Z. 246: 177--202, doi:10.1007/s00209-003-0594-z, ISSN 0025-5874, http://dx.doi.org/10.1007/s00209-003-0594-z 
  7. Silvestre, Luis (2006), [http://dx.doi.org/10.1512/iumj.2006.55.2706 "Hölder estimates for solutions of integro-differential equations like the fractional Laplace"], Indiana University Mathematics Journal 55 (3): 1155–1174, doi:10.1512/iumj.2006.55.2706, ISSN 0022-2518, http://dx.doi.org/10.1512/iumj.2006.55.2706 
  8. Caffarelli, Luis; Silvestre, Luis (2009), [http://dx.doi.org/10.1002/cpa.20274 "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 
  9. Silvestre, Luis (2011), "On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion", Advances in mathematics 226: 2020--2039 
  10. Lara, Héctor Chang; Dávila, Gonzalo (2012), Regularity for solutions of nonlocal, nonsymmetric equations, 29, pp. 833--859 
  11. Lara, Héctor Chang; Dávila, Gonzalo (2014), "Regularity for solutions of non local parabolic equations", Calculus of Variations and Partial Differential Equations 49: 139--172 
  12. Chang-Lara, Hector; Davila, Gonzalo, "H$\backslash$" older estimates for non-local parabolic equations with critical drift", arXiv preprint arXiv:1408.0676 
  13. Bjorland, C.; Caffarelli, L.; Figalli, A. (2012), "Non-local gradient dependent operators", Adv. Math. 230: 1859--1894, doi:10.1016/j.aim.2012.03.032, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2012.03.032 
  14. Schwab, Russell W; Silvestre, Luis, "Regularity for parabolic integro-differential equations with very irregular kernels", arXiv preprint arXiv:1412.3790 
  15. Kassmann, Moritz; Mimica, Ante, "Intrinsic scaling properties for nonlocal operators", arXiv preprint arXiv:1310.5371 
  16. Zacher, Rico (2013), "A De Giorgi--Nash type theorem for time fractional diffusion equations", Math. Ann. 356: 99--146, doi:10.1007/s00208-012-0834-9, ISSN 0025-5831, http://dx.doi.org/10.1007/s00208-012-0834-9 
  17. Barles, Guy; Chasseigne, Emmanuel; Imbert, Cyril (2011), "H\"older continuity of solutions of second-order non-linear elliptic integro-differential equations", J. Eur. Math. Soc. (JEMS) 13: 1--26, doi:10.4171/JEMS/242, ISSN 1435-9855, http://dx.doi.org/10.4171/JEMS/242 

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