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 $u_e \in L^\infty(\R^n)$.

Examples for which the estimate holds

Translation invariant, uniformly elliptic of order $s$, and some smoothness in the tails of the kernels

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 for some $\rho_0$ small enough. \begin{align*} \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\ \int_{\R^n \setminus B_{\rho_0}} \frac{|K(y)-K(y-h)|}{|h|} \mathrm d y &\leq C \qquad \text{every time $|h|<\frac {\rho_0} 2$} && \text{(kernel tails in $W^{1,1}$)} \end{align*}

Variant if the kernel tails are $C^1$

A small variation of the previous result is to assume the class of kernels satisfying the slightly stronger assumptions. A scale invariant class for which interior $C^{1,\alpha}$ regularity holds is \begin{align*} \frac{(2-s)\lambda}{|y|^{n+s}} \leq K(y) &\leq \frac{(2-s)\Lambda}{|y|^{n+s}} && \text{(standard unif. ellipticity of order $s$)}\\ \nabla K(y) &\leq \frac{\Lambda}{|y|^{n+s+1}} && \text{appropriate decay of the kernel in $C^1$.} \end{align*}

Then, any solution of $Iu=0$ in $B_r$ satisfies the estimate \[ [u]_{C^{1,\alpha}(B_{r/2})} \leq C \left(\frac 1 {r^{1+\alpha}} ||u||_{L^\infty(B_r)} + \frac 1 {r^{1+\alpha-s}} \int_{\R^n \setminus B_r} \frac{|u(y)|}{|y|^{n+s}} \mathrm d y \right). \] Other $C^{1,\alpha}$ estimates are obtained from this one using perturbation methods .

A class of non-differentiable kernels

A scale invariant class for which interior $C^{1,\alpha}$ regularity holds and the kernels can be very irregular is given by the following hypothesis \begin{align*} K(y) &= (2-s) \frac{a_1(y) + a_2(y)}{|y|^{n+s}} \\ \lambda &\leq a_1(y) \leq \Lambda \\ |a_2| &\leq \eta \\ |\nabla a_1(y)| &\leq \frac{C_1}{|y|} \qquad \text{in } \R^n \setminus \{0\} \end{align*} for $s>1$ and $\eta$ small enough (depending on $\lambda$, $\Lambda$, $C_1$ and dimension)

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*}