imported>Luis |
imported>Luis |
| Line 1: |
Line 1: |
| Hölder continuity of the solutions can sometimes be proved only from ellipticity
| | The linear integro-differential operators that we consider ''in this wiki'' are the generators of [[Levy processes]]. According to the Levy-Kintchine formula, they have the general form |
| 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 <ref name="rang2013h" /> <ref name="bogdan2005harnack" />.
| | \[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \] |
| | where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying |
| | \[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \] |
|
| |
|
| There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
| | The above definition is very general. In most cases we are interested in some subclass of linear operators. The simplest of all is the [[fractional Laplacian]]. We list several common simplifications below. |
| 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
| | == Absolutely continuous measure == |
| $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
| |
| (small). It is very important to allow 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 useful.
| |
|
| |
|
| In the non variational setting the integro-differential operators $L_x$ are | | In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$. |
| 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
| |
| $$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
| |
| K(x,y) \, dy$$
| |
| 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
| | We keep this assumption in all the examples below. |
| 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 == | | == Purely integro-differential operator == |
|
| |
|
| === Elliptic form ===
| | In this case we neglect the local part of the operator |
| 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
| | \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \] |
| \[ | |
| L(u,x) = 0 \ \ \text{in } B_1,
| |
| \]
| |
| 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.
| | == Symmetric kernels == |
| | If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient. |
|
| |
|
| For non variational problems, in order to adapt the situation to the [[viscosity solution]] framework, the equation may be replaced by two inequalities.
| | In the purely integro-differentiable case, it reads as |
| \begin{align*} | | \[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \] |
| 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 second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$. |
| 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 $\alpha > 0$,
| |
| \[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]
| |
|
| |
|
| | == Translation invariant operators == |
| | In this case, all coefficients are independent of $x$. |
| | \[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \] |
|
| |
|
| == Estimates which blow up as the order goes to two == | | == The fractional Laplacian == |
|
| |
|
| === Non variational case ===
| | The [[fractional laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous. |
|
| |
|
| The Hölder estimates were first obtained using probabilistic techniques <ref
| | \[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \] |
| name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref
| |
| name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to
| |
| a family satisfying certain set of assumptions. No regularity of any kind is
| |
| assumed for $K$ with respect to $x$. The assumption for the family of operators
| |
| are
| |
| # '''Scaling''': If $L$ belongs to the family, then so does its scaled version
| |
| $L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which
| |
| could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
| |
| # '''Nondegeneracy''': If $K$ is the kernel associated to $L$,
| |
| $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\inf_{B_1} K} \leq C_1$ for
| |
| some $C_1$ and $\alpha>0$ independent of $K$.
| |
|
| |
|
| The right hand side $f$ is assumed to belong to $L^\infty$.
| | == Uniformly elliptic of order $s$ == |
|
| |
|
| A particular case in which this result applies is the uniformly elliptic case.
| | This corresponds to the assumption |
| $$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
| | \[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \] |
| where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no
| |
| continuity of $s$ respect to $x$ is required.
| |
| The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$.
| |
| However this assumption can be overcome in the following two situations.
| |
| * For $s<1$, the symmetry assumption can be removed if the equation does not
| |
| contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy =
| |
| f(x)$ in $B_1$.
| |
| * For $s>1$, the symmetry assumption can be removed if the drift correction term
| |
| is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy =
| |
| f(x)$ in $B_1$.
| |
|
| |
|
| The reason for the symmetry assumption, or the modification of the drift | | The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two. |
| correction term, is that in the original formulation the term $y \cdot \nabla
| |
| u(x) \, \chi_{B_1}(y)$ is not scale invariant.
| |
|
| |
|
| === Variational case === | | == Order strictly below one == |
|
| |
|
| A typical example of a symmetric nonlocal [[Dirichlet form]] is a bilinear form
| | If a non symmetric kernel $K$ satisfies the extra local integrability assumption |
| $E(u,v)$ satisfying | | \[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \] |
| $$ E(u,v) = \iint_{\R^n \times \R^n} (v(y)-u(x))(v(y)-v(x)) K(x,y) \, dx
| | then the extra gradient term is not necessary in order to define the operator. |
| \, dy $$ | |
| 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
| | \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \] |
| 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
| | The modification in the integro-differential part of the operator becomes an extra drift term. |
| 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]].
| |
|
| |
|
| At some point in the original proof of De Giorgi, it is used that the
| | A uniformly elliptic operator of order $s<1$ satisfies this condition. |
| 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.
| |
|
| |
|
| == Estimates which pass to the second order limit == | | == Order strictly above one == |
|
| |
|
| === Non variational case ===
| | If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail. |
| | \[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \] |
| | then the gradient term in the integral can be taken global instead of being cut off in the unit ball. |
|
| |
|
| An integro-differential generalization of [[Krylov-Safonov]] theorem is
| | \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \] |
| available both in the elliptic <ref name="CS"/> and parabolic <ref name="lara2011regularity"/> setting. The assumption on the kernels are
| |
| # '''Symmetry''': $K(x,y) = K(x,-y)$.
| |
| # '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq
| |
| \frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$. | |
|
| |
|
| The right hand side $f$ is assumed to be in $L^\infty$. The constants in the | | The modification in the integro-differential part of the operator becomes an extra drift term. |
| Hölder estimate do not blow up as $s \to 2$.
| |
|
| |
|
| === Variational case ===
| | A uniformly elliptic operator of order $s>1$ satisfies this condition. |
| | |
| 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"/>.
| |
| | |
| == Other variants ==
| |
| | |
| * 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" />.
| |
| | |
| | |
| == References ==
| |
| {{reflist|refs=
| |
| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
| |
| estimates for solutions of integro-differential equations like the fractional
| |
| 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 |
| |
| 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
| |
| 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>
| |
| }}
| |
The linear integro-differential operators that we consider in this wiki are the generators of Levy processes. According to the Levy-Kintchine formula, they have the general form
\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
The above definition is very general. In most cases we are interested in some subclass of linear operators. The simplest of all is the fractional Laplacian. We list several common simplifications below.
Absolutely continuous measure
In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
We keep this assumption in all the examples below.
Purely integro-differential operator
In this case we neglect the local part of the operator
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]
Symmetric kernels
If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.
In the purely integro-differentiable case, it reads as
\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]
The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
Translation invariant operators
In this case, all coefficients are independent of $x$.
\[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]
The fractional Laplacian
The fractional laplacian is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.
\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]
Uniformly elliptic of order $s$
This corresponds to the assumption
\[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]
The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
Order strictly below one
If a non symmetric kernel $K$ satisfies the extra local integrability assumption
\[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
then the extra gradient term is not necessary in order to define the operator.
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]
The modification in the integro-differential part of the operator becomes an extra drift term.
A uniformly elliptic operator of order $s<1$ satisfies this condition.
Order strictly above one
If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
\[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
then the gradient term in the integral can be taken global instead of being cut off in the unit ball.
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]
The modification in the integro-differential part of the operator becomes an extra drift term.
A uniformly elliptic operator of order $s>1$ satisfies this condition.