Differentiability estimates

Given a fully nonlinear integro-differential equation $Iu=0$, uniformly elliptic with respect to certain class of operators, sometimes an interior $C^{1,\alpha}$ estimate holds for some $\alpha>0$ (typically very small). Assume $I0=0$. The $C^{1,\alpha}$ estimate is a result like the following.

Theorem. Let $u \in L^\infty(R^n) \cap C(\overline B_1)$ solve the equation \[Iu = 0 \ \ \text{in } B_1.\] Then $u \in C^{1,\alpha}(B_{1/2})$ and the following estimate holds \[ ||u||_{C^{1,\alpha}(B_{1/2})} \leq C ||u||_{L^\infty}. \]

A theorem as above is known to hold under some assumptions on the nonlocal operator $I$. A list of valid assumptions is provided below.

Note that the result is stated for general fully nonlinear integro-differential equations, but the most important cases to apply it are the Isaacs equation and Bellman equation.

Idea of the proof

The idea to prove a $C^{1,\alpha}$ estimate is to apply Holder estimates to the derivatives of the solutions $u$. The directional derivatives $u_e$ satisfy the two inequalities \[ M^+_{\mathcal L} u_e \geq 0 \text{ and } M^-_{\mathcal L} u_e \leq 0 \] where $M^\pm_{\mathcal L}$ are the extremal operators with respect to the corresponding class of operators $\mathcal L$. If the Holder estimates apply to this class of operators, one would expect that $u_e \in C^\alpha$ for any vector $e$, and therefore $u \in C^{1,\alpha}$.

There is a technical problem with the idea above. The Holder estimates indicate that $u_e$ is $C^\alpha$ in some $B_{1/2}$ provided that $u_e$ is already known to be bounded in $L^\infty(\R^n)$. In order to obtain the estimate starting from $u \in L^\infty(\R^n)$, one applies the Holder estimates successively to gain regularity at every step and then prove iteratively that $u \in C^\alpha \Rightarrow u \in C^{2\alpha} \Rightarrow u \in C^{3\alpha} \Rightarrow \dots \Rightarrow u \in C^{1,\alpha}$. The last step in the iteration illustrates the difficulty. Imagine that we have already proved that $u$ is Lipschitz in $B_{3/4}$, so we know that $u_e \in L^\infty(B_{3/4})$ for any vector $e$. This is not enough to apply the Holder estimates to $u_e$ since we would need $u_e \in L^\infty(\R^n)$. Because of this difficulty, the first versions of the proof assume an extra regularity condition on the family of kernels. This regularity condition can be removed following the methods in and .

Examples for which the estimate holds

Translation invariant, uniformly elliptic of order $s$

The first situation in which the interior $C^{1,\alpha}$ estimate was proved for a nonlocal equation was if $I$ is translation invariant and uniformly elliptic with respect to the class of kernels satisfying the following hypothesis. \[ \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) \leq \frac{(2-s)\Lambda}{|y|^{n+s}} \ \ \text{(standard unif. ellipticity of order $s$)} \]

The result was first proved in assuming an extra regularity condition in the family of kernels. This condition was later removed in . For the parabolic version of the problem, it was first done in with the extra smoothness assumption on the kernel, which was later removed in .

Isaacs equation with variable coefficients but close to constant

If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates . The family of integro-differential operators has kernels which are the sum of a fixed term $a_0$ (the same for all kernels in the class) and a small term which can depend on $x$. \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)(a_0(y) + a_{\alpha \beta}(x,y))}{|y|^{n+s}} \mathrm d y =0\] such that we have for $\eta$ small enough and any $\alpha$, $\beta$, \begin{align*} |a_{\alpha \beta}(x,y)| &< \eta \qquad \text{ for every } \alpha, \beta \\ \lambda &\leq a_0(y) \leq \Lambda \\ |\nabla a_0(y)| &\leq C |y|^{-1} \end{align*} (note that this $C^{1,\alpha}$ estimate is nontrivial in the linear case as well)

Isaacs equation with continuous coefficients

If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates . \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\] such that for every $\alpha$, $\beta$ we have \begin{align*} \lambda \leq a_{\alpha \beta}(x,y) &\leq \Lambda \\ \nabla_y a_{\alpha \beta}(x,y) &\leq C_1/((2-s)|y|)\\ |a_{\alpha \beta}(x_1,y) - a_{\alpha \beta}(x_2,y)| &\leq c(|x_1-x_2|) && \text{for some uniform modulus of continuity $c$}. \end{align*}