Differentiability estimates

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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.

Contents

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 [1] and [2].

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 [3] assuming an extra regularity condition in the family of kernels. This condition was later removed in [1]. For the parabolic version of the problem, it was first done in [4] with the extra smoothness assumption on the kernel, which was later removed in [2].

Isaacs equation with variable coefficients but close to constant

If $s>1$, the following Isaacs equation also has interior $C^{1,\alpha}$ estimates [5]. 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 [5]. \[ \inf_\alpha \ \sup_\beta \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \frac{(2-s)a_{\alpha \beta}(x,y)}{|y|^{n+s}} \mathrm d y = 0\] 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*}


References

  1. 1.0 1.1 Kriventsov, Dennis (2013), "C 1, $\alpha$ Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels", Communications in Partial Differential Equations 38: 2081--2106 
  2. 2.0 2.1 Serra, Joaquim, "Regularity for fully nonlinear nonlocal parabolic equations with rough kernels", arXiv preprint arXiv:1401.4521 
  3. Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274 
  4. Lara, HéctorChang; Dávila, Gonzalo, "Regularity for solutions of non local parabolic equations", Calculus of Variations and Partial Differential Equations (Springer Berlin Heidelberg) 49: 139-172, doi:10.1007/s00526-012-0576-2, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-012-0576-2 
  5. 5.0 5.1 Caffarelli, Luis; Silvestre, Luis (2009), "Regularity results for nonlocal equations by approximation", Archive for Rational Mechanics and Analysis (Berlin, New York: Springer-Verlag): 1–30, ISSN 0003-9527 
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