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| Hölder continuity of the solutions can sometimes be proved only from ellipticity
| | In broad and vague terms, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that minimize (within a class of given admissible configurations) the energy functional: |
| 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]].
| | \[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \] |
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| There are integro-differential versions of both [[De Giorgi-Nash-Moser theorem]]
| | It can be checked easily that this agrees (save for a factor of $2$) with norm of the characteristic function $\chi_E$ in the homogenous Sobolev space $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$. |
| 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
| | Classically, [[minimal surfaces]] (or generally [[surfaces of constant mean curvature]] ) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart. |
| $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$
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| (small). It is very important to allow 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 useful.
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|
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| In the non variational setting the integro-differential operators $L_x$ are
| | Nonlocal minimal surfaces then, describe physical phenomena where the interaction potential does not decay fast enough as particles are apart, so that two particles on different phases and far from the interface still contribute a non-trivial amount to the total interaction energy, in particular, one may consider much more general energy functionals corresponding to different interaction potentials |
| 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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| $$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y))
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| K(x,y) \, dy$$
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| Since [[linear integro-differential operators]] allow for a great flexibility of
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| equations, there are several variations on the result: different assumptions on
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| 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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| As with other estimates in this field too, some Hölder estimates blow up as the
| | \[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \] |
| 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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| == The general statement == | | == Definition == |
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| === Elliptic form ===
| | Following the most accepted convention for [[minimal surfaces]], a nonlocal minimal surface is the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\chi_E \in \dot{H}^{s/2}$ and whose [[Nonlocal mean curvature]] $H_s$ is identically zero (see next section), that is |
| 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.
| | \[ H_s(x): = C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\] |
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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.
| | In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$. |
| \begin{align*} | |
| 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 $\alpha > 0$,
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| \[ \|u\|_{C^\alpha((-1/2,0] \times B_{1/2})} \leq C \|u\|_{L^\infty((-1,0] \times \R^d)}.\]
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| | Example: Suppose that $E$ and $\Omega$ are such that for any other set $F$ such that $F \Delta E \subset \subset \Omega$ (i.e. $F$ agrees with $E$ outside $\Omega$) we have |
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| == Estimates which blow up as the order goes to two ==
| | \[J_s(E) \leq J_s(F) \] |
|
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| === Non variational case ===
| | Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$. |
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| The Hölder estimates were first obtained using probabilistic techniques <ref
| | <div style="background:#DDEEFF;"> |
| name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref
| | <blockquote> |
| name="S"/>. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to
| | '''Note''' For this definition to make sense, $\Sigma$ must be the boundary of some open set $E$, in this article, we will often refer to the set $E$ itself as "the" minimal surface, and no confusion should arise from this. |
| a family satisfying certain set of assumptions. No regularity of any kind is
| | </blockquote> |
| assumed for $K$ with respect to $x$. The assumption for the family of operators
| | </div> |
| are
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| # '''Scaling''': If $L$ belongs to the family, then so does its scaled version
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| $L_r u(x) = C_{r,L} L [u(x/r)] (x)$ 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 $L$.
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| # '''Nondegeneracy''': If $K$ is the kernel associated to $L$,
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| $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\inf_{B_1} K} \leq C_1$ for
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| some $C_1$ and $\alpha>0$ independent of $K$.
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| The right hand side $f$ is assumed to belong to $L^\infty$.
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| A particular case in which this result applies is the uniformly elliptic case.
| | == Nonlocal mean curvature == |
| $$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
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| where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no
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| continuity of $s$ respect to $x$ is required.
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| The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$.
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| However this assumption can be overcome in the following two situations.
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| * For $s<1$, the symmetry assumption can be removed if the equation does not
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| contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy =
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| f(x)$ in $B_1$.
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| * For $s>1$, the symmetry assumption can be removed if the drift correction term
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| is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy =
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| f(x)$ in $B_1$.
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|
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| The reason for the symmetry assumption, or the modification of the drift
| | == Surfaces minimizing non-local energy functionals == |
| correction term, is that in the original formulation the term $y \cdot \nabla
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| u(x) \, \chi_{B_1}(y)$ is not scale invariant.
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|
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| === Variational case === | | == The Caffarelli-Roquejoffre-Savin Regularity Theorem== |
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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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| $$ \lim_{\eps \to 0} \int_{|x-y|>\eps} (u(y) - u(x) ) K(x,y) \, dy = 0,$$
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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
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| 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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| == Estimates which pass to the second order limit ==
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| === Non variational case ===
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| An integro-differential generalization of [[Krylov-Safonov]] theorem is
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| available both in the elliptic <ref name="CS"/> and parabolic <ref name="lara2011regularity"/> setting. The assumption on the kernels are
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| # '''Symmetry''': $K(x,y) = K(x,-y)$.
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| # '''Uniform ellipticity''': $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq
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| \frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.
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| The right hand side $f$ is assumed to be in $L^\infty$. The constants in the
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| Hölder estimate do not blow up as $s \to 2$.
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| === Variational case ===
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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"/>.
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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 very general class of integral equations <ref name="barles2011" />.
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| == References ==
| |
| {{reflist|refs=
| |
| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
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| 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 |
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| pages=1155–1174}}</ref>
| |
| <ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre |
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| 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]] |
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| issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
| |
| <ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan |
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| first2=Chi Hin | last3=Vasseur | first3=Alexis | title= |
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| 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 |
| |
| pages=1249–1259}}</ref>
| |
| <ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin |
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| 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
| |
| Functional Analysis | issn=0926-2601 | volume=17 | issue=4 |
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| 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>
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| }}
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