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| Hölder continuity of the solutions can sometimes be proved only from ellipticity
| | The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely |
| assumptions on the equation, without depending on smoothness of the
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| coefficients. This allows great flexibility in terms of applications of the
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| result. The corresponding result for elliptic equations of second order is the
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| [[Krylov-Safonov]] theorem in the non-divergence form, or the [[De Giorgi-Nash-Moser theorem]] in the divergence form.
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| 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" />.
| | \[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\] |
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| There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
| | \[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \] |
| and [[Krylov-Safonov theorem]]. The former uses variational techniques and is
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| stated in terms of Dirichlet forms. The latter is based on comparison
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| principles.
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| A Hölder estimate says that a solution to an integro-differential equation with rough coefficients
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| $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
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| (small). It is very important when an estimate allows for a very rough dependence of
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| $L_x$ with respect to $x$, since the result then applies to the linearization of
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| (fully) nonlinear equations without any extra a priori estimate. On the other
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| 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
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| respect to $y$. Therefore, smoothness (or even structural) assumptions for the
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| kernels with respect to $y$ can be made keeping such result applicable.
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|
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| In the non variational setting the integro-differential operators $L_x$ are
| | \[ u_t +(-\Delta)^{s}(u^m) = 0 \] |
| assumed to belong to some family, but no continuity is assumed for its
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| dependence with respect to $x$. Typically, $L_x u(x)$ has the form
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| $$ L_x u(x) = a_{ij}(x) \partial_{ij} u + b(x) \cdot \nabla u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
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| K(x,y) \, dy$$
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| 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.
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|
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| Since [[linear integro-differential operators]] allow for a great flexibility of
| | These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order [[Quasilinear equations]]. |
| equations, there are several variations on the result: different assumptions on | |
| the kernels, mixed local terms, evolution equations, etc. The linear equation
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| with rough coefficients is equivalent to the function $u$ satisfying two
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| inequalities for the [[extremal operators]] corresponding to the family of
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| operators $L$, which stresses the nonlinear character of the estimates.
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|
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| As with other estimates in this field too, some Hölder estimates blow up as the
| | The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which [[instantaneous speed of propagation]] holds. |
| order of the equation converges to two, and others pass to the limit. The
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| blow-up is a matter of the techniques used in the proof. Only estimates which
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| are robust are a true generalization of either the [[De Giorgi-Nash-Moser theorem]] or
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| [[Krylov-Safonov theorem]].
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|
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| == The general statement == | | This means that the first model presents us with a [[free boundary problem]]. For this model global existence and Hölder continuity of weak solutions have been recently obtained <ref name="CV1"/>, there is almost nothing known about the properties of its free boundary, making it a rich source of open questions. |
|
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| === Elliptic form ===
| | For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years <ref name="PQRV" /ref>. |
| 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
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| \[
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| L(u,x) = 0 \ \ \text{in } B_1,
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| \]
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| and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$,
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| \[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^d)}.\]
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| 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.
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| 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*}
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| M^+u \geq 0 \ \ \text{in } B_1, \\
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| M^-u \leq 0 \ \ \text{in } B_1.
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| \end{align*}
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| where $M^+$ and $M^-$ are [[extremal operators]] with respect to some class.
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| | |
| === Parabolic form ===
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| 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
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| \[
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| u_t - L(u,x) = 0 \ \ \text{in } (-1,0] \times B_1,
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| \]
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| and $u \in L^\infty(\R^d)$, then for some small $\alpha > 0$, | |
| \[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]
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| == List of results ==
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| 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.
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| === Variational equations ===
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| A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
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| $E(u,v)$ satisfying
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| $$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx
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| \, dy $$
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| on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note
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| that $K$ can be assumed to be symmetric because the skew-symmetric part
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| of $K$ would be ignored by the bilinear form.
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| Minimizers of the corresponding quadratic forms satisfy the nonlocal Euler
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| equation
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| $$ \lim_{\varepsilon \to 0} \int_{|x-y|>\varepsilon} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
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| which should be understood in the sense of distributions.
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| It is known that the gradient flow of a Dirichlet form (parabolic version of the
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| result) becomes instantaneously Hölder continuous <ref name="CCV"/>. The method
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| of the proof builds an integro-differential version of the parabolic De Giorgi
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| technique that was developed for the study of critical [[Surface quasi-geostrophic equation]].
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| At some point in the original proof of De Giorgi, it is used that the
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| characteristic functions of a set of positive measure do not belong to $H^1$.
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| Moreover, a quantitative estimate is required about the measure of
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| ''intermediate'' level sets for $H^1$ functions. In the integro-differential
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| context, the required statement to carry out the proof would be the same with
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| the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would
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| even require a non trivial proof for $s$ close to $2$. The difficulty is
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| bypassed though an argument that takes advantage of the nonlocal character of
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| the equation, and hence the estimate blows up as the order approaches two.
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| In the stationary case, it is known that minimizers of Dirichlet forms are
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| Hölder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser theorem]] to the
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| nonlocal setting <ref name="K"/>. In this result, the constants do not blow up as the order of the equation goes to two.
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| {{stub}}
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| === Non variational equations ===
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| The results below are nonlocal versions of [Krylov-Safonov theorem]. No regularity needs to be
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| assumed for $K$ with respect to $x$.
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| | |
| * The first result was obtained by Bass and Levin using probabilistic techniques.<ref
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| 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" />.
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| The estimate says the following. Assume a bounded function $u: \R^n \to \R$ solves
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| $$ \int_{\R^n} (u(x+y) - u(x)) K(x,y) \mathrm{d}y = 0 \qquad \text{for all $x$ in } B_1, $$
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| where $K$ satisfies the symmetry assumption $K(x,y) = K(x,-y)$ and
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| \begin{equation} \label{pointwisebound}
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| \frac{\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s}} \qquad \text{for all } x,y \in \R^n,
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| \end{equation}
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| where $s \in (0,2)$ and $\Lambda \geq \lambda > 0$ are given parameters. Then
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| \[ \|u\|_{C^\alpha(B_{1/2})} \leq C \|u\|_{L^\infty(\R^n)}.\]
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| | |
| * A result by Bass and Kassmann also uses probabilistic techniques.<ref
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| name="BK"/> 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.
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| * The first purely analytic proof was obtained by Luis Silvestre <ref
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| 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.
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| * 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}).
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| * 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.
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| * 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}).
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| * 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"/>.
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| * 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.
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| == Other variants ==
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| * There are Holder estimates for equations in divergence form that are non local in time <ref name="zacher2013" />
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| * If we allow for continuous dependence on the coefficients with respect to $x$, there are Hölder estimates for a rather singular class of integral equations <ref name="barles2011" />.
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|
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|
| == References == | | == References == |
| {{reflist|refs= | | {{reflist|refs= |
| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder | | <ref name="CV1"> {{Citation | last1=Caffarelli | first1=Luis | last2=Vazquez | first2=Juan | title=Nonlinear Porous Medium Flow with Fractional Potential Pressure | url=http://dx.doi.org/10.1007/s00205-011-0420-4 | publisher=Springer Berlin / Heidelberg | year=2011 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–29}} </ref> |
| estimates for solutions of integro-differential equations like the fractional
| | |
| Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
| | <ref name="PQRV">{{Citation | last1=Pablo | first1=Arturo de | last2=Quirós | first2=Fernando | last3=Rodríguez | first3=Ana | last4=Vazquez | first4=Juan Luis | title=A fractional porous medium equation | url=http://www.sciencedirect.com/science/article/pii/S0001870810003130 | doi=DOI: 10.1016/j.aim.2010.07.017 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1378–1409}} |
| doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
| | </ref> |
| Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
| |
| pages=1155–1174}}</ref>
| |
| <ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre |
| |
| 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
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| American Mathematical Society]] | issn=0894-0347 | issue=24 |
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| pages=849–869}}</ref>
| |
| <ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori
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| 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
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| Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
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| 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
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| Interactions between Potential Theory, Probability Theory, Geometry and
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| Functional Analysis | issn=0926-2601 | volume=17 | issue=4 |
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| pages=375–388}}</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>
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| }} | | }} |
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| | [[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]] |