Hölder estimates: Difference between revisions

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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}).
* 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.
* 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.
* An estimate for elliptic equations with non symmetric kernels was obtained by Davila and Chang-Lara <ref name="lara2012regularity" />. In their first result, the odd part of the kernels was supposed to be of lower order compared to their symmetric part. This requirement was removed in subsequent work <ref name="chang2014h" />. The kernels are required to be uniformly elliptic (pointwise bounded as in \ref{pointwisebound}).
* 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]].


== Estimates which blow up as the order goes to two ==
== Estimates which blow up as the order goes to two ==
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<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="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="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 16:09, 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 o that, the coefficients $a_{ij}(x)$ are usually 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

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.[3] 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 [4].

The estimate says the following. Assume a bounded function $u: \R^n \to \R$ solves $$ \int_{\R^n} (u(x+y) - u(x)) K(x,y) \mathrm{d}y = 0 \qquad \text{in } B_1, $$ where $K$ satisfies \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 B_1, \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.[3] 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 [5]. 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 [6]. 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 [7]. 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 [8]. A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.[9]. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels [10]. 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 [11] 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 [12]. 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.

Estimates which blow up as the order goes to two

Non variational case

The Hölder estimates were first obtained using probabilistic techniques [3] [13] , and then using purely analytic methods [5]. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to a family satisfying certain set of assumptions. No regularity needs to be assumed for $K$ with respect to $x$. The assumption for the family of operators are

  1. 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 $L$.

  1. 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 $L^\infty$.

A particular case in which this result applies is the uniformly elliptic case. $$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$ 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 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

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 [14]. 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.

Estimates which pass to the second order limit

Non variational case

An integro-differential generalization of Krylov-Safonov theorem is available both in the elliptic [6] and parabolic [15] setting. The assumption on the kernels are

  1. Symmetry: $K(x,y) = K(x,-y)$.
  2. 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 Hölder estimate do not blow up as $s \to 2$.

Variational case

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 [16].

Other variants

  • There are Holder estimates for equations in divergence form that are non local in time [17]
  • 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 [18].


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. 3.0 3.1 3.2 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 
  4. 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 
  5. 5.0 5.1 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 
  6. 6.0 6.1 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 
  7. Silvestre, Luis (2011), "On the differentiability of the solution to the Hamilton--Jacobi equation with critical fractional diffusion", Advances in mathematics 226: 2020--2039 
  8. Lara, Héctor Chang; Dávila, Gonzalo (2012), Regularity for solutions of nonlocal, nonsymmetric equations, 29, pp. 833--859 
  9. Lara, Héctor Chang; Dávila, Gonzalo (2014), "Regularity for solutions of non local parabolic equations", Calculus of Variations and Partial Differential Equations 49: 139--172 
  10. Chang-Lara, Hector; Davila, Gonzalo, "H$\backslash$" older estimates for non-local parabolic equations with critical drift", arXiv preprint arXiv:1408.0676 
  11. 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 
  12. Schwab, Russell W; Silvestre, Luis, "Regularity for parabolic integro-differential equations with very irregular kernels", arXiv preprint arXiv:1412.3790 
  13. Bass, Richard F.; Kassmann, Moritz (2005), [http://dx.doi.org/10.1080/03605300500257677 "Hölder continuity of harmonic functions with respect to operators of variable order"], Communications in Partial Differential Equations 30 (7): 1249–1259, doi:10.1080/03605300500257677, ISSN 0360-5302, http://dx.doi.org/10.1080/03605300500257677 
  14. 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 
  15. Lara, Héctor Chang; Dávila, Gonzalo (2011), "Regularity for solutions of non local parabolic equations", Calculus of Variations and Partial Differential Equations: 1--34 
  16. 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 
  17. 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 
  18. 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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