Hölder estimates

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 .

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 of that, the coefficients $a_{ij}(x)$ are usually assumed to be 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

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 . 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.

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 . In this result, the constants do not blow up as the order of the equation goes to two.

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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. 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 .

The estimate says the following. Assume a bounded function $u: \R^n \to \R$ solves $$ \int_{\R^n} (u(x+y) - u(x)) K(x,y) \mathrm{d}y = 0 \qquad \text{for all $x$ in } B_1, $$ where $K$ satisfies the symmetry assumption $K(x,y) = K(x,-y)$ and \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 \R^n, \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. 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 Silvestre . 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 . 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 . 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 . A later work provides estimates for parabolic equations which stay uniform as the order of the equation goes to two.. In a newer paper they improved both of their previous results by providing estimates for parabolic equations, with nonsymmetric kernels . 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 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 .
• A generalization of all previous results (except for variable order) was obtained by Schwab and Silvestre . 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.
• A non scale invariant family of kernels was studied by Kassmann and Mimica . 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. For some kernels of the form $K(x,y) \approx |y|^{-n}$, they obtain an estimate of a modulus of continuity for the solution $u$ which is not Holder.

Other variants

• There are Holder estimates for equations in divergence form that are nonlocal in time , and also nonlocal in space-time together .
• 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 .
• The interaction of an integral diffusion with first order terms brings some extra issues that are described in the page of drift-diffusion equations.