Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms

In this article, we examine the regularity theory of viscosity solutions to a fully nonlinear nonlocal equation of the form

where $\sigma\in(0,2)$, $C(\sigma)>0$ is a normalizing constant, $\delta(u,x,y):=u(x+y)+u(x-y)-2u(x)$, ${\mathcal{A}}$ and ${\mathcal{B}}$ are indexes sets, and the source term $f$ belongs to a suitable Lesbegue space. Moreover, we suppose the following ellipticity-type condition: for each $\alpha\in{\mathcal{A}}$ and $\beta\in{\mathcal{B}}$, $A_{\alpha,\beta}$ is a symmetric $d\times d$ matrix $(\lambda,\Lambda)$-elliptic, i.e.,

for $0<\lambda\leq\Lambda$. We establish Hölder regularity for solutions (or their gradient), depending on the range of $p$ for which $f\in L^{p}(B_{1})$.

Regularity results for nonlocal operators have been extensively studied over the years. In [15], the author presents an analytical proof of Hölder continuity and introduces more flexible assumptions on the operator than previous studies. But it is only in [6], where such regularity estimates were given uniformly as the degree of the operator $\sigma\to 2$, seen as a natural extension of second-order operators. In that paper, the authors also prove properties such as comparison principle, a nonlocal Alexandrov-Bakelman-Pucci estimate (ABP for short), Harnack inequality, and regularity in $C^{1,\alpha}$ spaces for viscosity solutions to equations of type (1.1).

Numerous results were uncovered based on the findings in [6]. In [8], the authors extended their previous results using perturbative methods, including the $C^{1,\alpha}$-regularity for a class of non translation-invariant equations. Around the same time, the authors in [3] also worked with non translation-invariant equations, establishing Hölder regularity of solutions, using different assumptions from [8], allowing first-order terms and some degeneracy in the operators. We also mention the work [12], where the author gives $C^{1,\alpha}$-estimates for more general kernels than previous results.

Besides the $C^{1,\alpha}$ estimates, Evans-Krylov-type results have been explored by the community. In this direction, the authors in [7] proved that viscosity solutions for concave (or convex) equations of the form

are of class $C^{\sigma+\beta}$. They assume that, for each $\alpha\in{\mathcal{A}}$, the kernel $K_{\alpha}$ belongs to the class ${\mathcal{L}}_{2}$, i.e., they are of class $C^{2}$ away from the origin, symmetric, satisfies the ellipticity condition

and

In [13], the author extended the result above to equations of type (1.3) with rough kernels, i.e., where the kernels $K_{\alpha}$ satisfy (1.4) but not necessarily (1.5), for every $\alpha\in{\mathcal{A}}$. Under the additional (and optimal) assumption of $C^{\alpha}$ exterior data for the solution, the author provides $C^{\sigma+\alpha}$ a priori estimates. Moreover, results involving non translation-invariant kernels are also given, provided the dependency on the variable $x$ is of a $C^{\alpha}$-fashion. We also mention the work [18], where the author also provides $C^{\sigma+\alpha}$ for viscosity solutions to nonlocal equations of the type

provided the quantities $\|u\|_{L^{\infty}(B_{1})}$ and $\|u\|_{L^{1}_{\sigma}(\mathbb{R}^{d})}$ are sufficiently small.

A natural question is whether one can prove optimal regularity estimates for fully nonlinear nonlocal equations with an unbounded right-hand side. However, such results are relatively scarce in the literature. One of the primary challenges in this context is the lack of compactness properties for viscosity solutions of (1.1) (or even the simpler equation (1.7) below) when $f\in L^{p}(B_{1})$. In addition, stability results are also crucial for implementing the now-classic strategy proposed in [8]. In fact, only a few works involving an unbounded norm on the right-hand side are available. Moreover, the existing results consider a class of kernels that differ from the broader ${\mathcal{L}}_{0}$ class discussed in [6, 8, 12, 13], which defines the extremal operator

Instead, they work with kernels of the form

where $A_{\alpha}$ is a symmetric $d\times d$ matrix satisfying

for positive constants $0<\lambda\leq\Lambda$. These kernels define a class of extremal operators as

It is important to note that, because they allow some degeneracy (conform (1.7)), although this is a smaller class, it is not necessarily contained within the ${\mathcal{M}}^{-}_{{\mathcal{L}}_{0}}$.

In this direction, the authors in [9] established a quantitative ABP estimate for viscosity supersolutions to

They show that supersolutions to the equation above satisfy

where $K_{u}$ is the coincidence set between $u$ and a type of fractional convex envelope of $u$. In [10], the author improved the previous result, by removing the dependency of the $L^{\infty}$-norm in the estimate above, in case $\sigma$ is sufficiently close to 2. In [11], the author removed the restriction on the degree of the operator present in [10], by proving that supersolutions to (1.8) with $f\in L^{p}(B_{1})$, for $p\in(d-\varepsilon_{0},\infty)$, satisfy

Moreover, the author also gives $W^{\sigma,p}$-estimates for viscosity solutions to concave equations of the form

which, in particular, implies that solutions are of class $C^{\sigma-d/p}$. To our knowledge, this is the only result in the literature that gives this type of regularity for fully nonlinear nonlocal equations as in (1.9), in the presence of an unbounded right-hand side.

The main purpose of this paper is to establish optimal regularity estimates for viscosity solutions to equation (1.1), where the source term $f\in L^{p}(B_{1})$. We emphasize that such estimates have not been previously available for this type of equation. We employ the so-called half-relaxed method, originally introduced in [4], to prove that a sequence $(u_{k})_{k\in\mathbb{N}}$ of merely bounded viscosity solutions to (1.1) converges uniformly to a function $u_{\infty}$ which solves a suitable equation, see Lemma 2.11 below. This method relies on the comparison principle of the operator and has been used previously in the context of Hamilton-Jacobi equations and nonlocal equations with Neumann boundary conditions in a half-space. We refer the reader to [1, 2]. Once compactness and stability are available, standard approximation arguments can be applied to show the regularity properties of the viscosity solutions.

As discussed earlier, the regularity of solutions depends on the range of $p$ for which $f\in L^{p}(B_{1})$, in the spirit of [16]. First, we prove that if $p\in(d-\varepsilon_{0},\frac{d}{\sigma-1})$, then solutions are of class $C^{\sigma-d/p}$. The constant $\varepsilon_{0}$ is known as the Escauriaza’s exponent. Notice that this type of regularity was previously known from [11] in the context of a concave equation as in (1.9).

The borderline case $f\in L^{d/(\sigma-1)}$ is particularly significant. In the local case, i.e., $\sigma=2$, this value separates continuity estimates from differentiability properties in the regularity theory. Moreover, this quantity also appears in ABP and Harnack estimates. Meanwhile, in the nonlocal case, this is the first time such a threshold has been explicitly considered, as previous ABP estimates in [9, 10, 11] only consider the $L^{d}$-norm of the right-hand side. We believe this is the correct threshold for future general ABP estimates for the general class ${\mathcal{L}}_{0}$. Nevertheless, in this scenario, we show that viscosity solutions to (1.1) are Log-Lipschitz, which is better than $C^{\alpha}$ for every $\alpha\in(0,1)$.

For $f\in L^{p}$ for $p\in(\frac{d}{\sigma-1},+\infty)$, we prove that the solutions belong to the class $C^{1,\alpha}$, where $\alpha$ is defined in (2.2). Finally, for the borderline case where $f\in\rm{BMO}(B_{1})$, we show that viscosity solutions to (1.9) are locally of class $C^{\sigma}$, which is the best regularity we can hope for without assuming further regularity for $f$. These results are detailed in Theorems 2.6, 2.7, 2.8 and 2.9 below.

The class of kernels in (1.1) follows the form of (1.6), as described in [9, 10, 11], but with the additional requirement of uniform ellipticity, as in (1.2). In particular, the class of kernels that we deal with is a subset of ${\mathcal{L}}_{0}$ and enjoys all its properties. Furthermore, because uniform ellipticity as in (1.2) ensures that the condition (1.7) is satisfied with the same ellipticity constants, our class of kernels is also included in those described in [9, 10, 11]. Developing a similar theory for more general kernels, such as those in ${\mathcal{L}}_{0}$, remains an open challenge, primarily due to the need for an appropriate ABP estimate, as discussed earlier.

The remainder of this paper is structured as follows: In section 2 we gather some auxiliary results and present our main results. The proof of the optimal Hölder regularity is the subject of Section 3. In Section 4 we put forward the Log-Lipschitz regularity of solutions. Section 5 is devoted to the proof of Hölder regularity for the gradient of solutions. Finally, in the last section, we investigate the borderline $C^{\sigma}$-regularity.