Hölder estimates and Holder estimates: Difference between pages

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Hölder continuity of the solutions can sometimes be proved only from ellipticity
#REDIRECT [[Hölder estimates]]
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" />.
 
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.<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" />.
 
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.<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). 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).
 
== Estimates which blow up as the order goes to two ==
 
=== Non variational case ===
 
The Hölder estimates were first obtained using probabilistic techniques <ref
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 needs to be
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 $L^\infty$.
 
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 <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
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 <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
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 <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
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<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
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pages=849–869}}</ref>
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<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>
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}}

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