[Paper Review] Boundary regularity for nonlocal operators with kernels of variable orders

Minhyun Kim, Panki Kim+2 more|arXiv (Cornell University)|Apr 5, 2018

Nonlinear Partial Differential Equations26 references3 citations

TL;DR

This paper establishes boundary regularity for viscosity solutions of nonlocal Dirichlet problems with variable-order kernels by introducing a generalized Hölder space framework. It proves that solutions satisfy $ u \in C^V(D) $ and $ u/V(d_D) \in C^\alpha(D) $ with uniform estimates, where $ V $ is the renewal function and $ d_D $ is the distance to the boundary, under weak scaling and decay conditions on the kernel's Lévy density.

ABSTRACT

We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Hölder space. We prove that there exists a unique viscosity solution of $Lu = f$ in $D$, $u=0$ in $\mathbb{R}^n \setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u \in C^V(D)$ and $u/V(d_D) \in C^α(D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.

Motivation & Objective

• To establish boundary regularity for viscosity solutions of nonlocal Dirichlet problems with kernels of variable order.
• To extend classical Hölder regularity theory to nonlocal operators where the order of differentiation varies spatially.
• To characterize the precise boundary behavior of solutions using the renewal function $ V $ associated with the underlying Lévy process.
• To prove uniform estimates for the solution $ u $ in terms of the distance function $ d_D $ and the renewal function $ V $.
• To generalize results from fractional Laplacians (constant order) to nonlocal operators with variable-order kernels under weak scaling and decay conditions.

Proposed method

• Use probabilistic tools by interpreting the nonlocal operator as the infinitesimal generator of a subordinate Brownian motion.
• Employ the Bernstein function $ \phi $ to model variable-order behavior, with $ \phi(\lambda) $ satisfying weak scaling at infinity.
• Define the renewal function $ V $ associated with the one-dimensional Lévy process, which controls the boundary behavior of solutions.
• Construct barriers and use Harnack-type inequalities to control oscillation of $ u/V(d_D) $ near the boundary.
• Apply interpolation and oscillation estimates in dyadic annuli to prove Hölder continuity of $ u/V(d_D) $ in $ D $.
• Use the potential operator for the killed process of subordinate Brownian motion to derive a priori estimates.

Experimental results

Research questions

• RQ1 How does the solution $ u $ of the nonlocal Dirichlet problem behave near the boundary when the kernel has variable order?
• RQ2 What role does the renewal function $ V $ play in characterizing the boundary regularity of solutions?
• RQ3 Can the classical Hölder regularity theory be extended to nonlocal operators with variable-order kernels?
• RQ4 Under what conditions on the Lévy density does the solution $ u $ satisfy $ u/V(d_D) \in C^\alpha(D) $ with uniform estimates?
• RQ5 How do weak scaling and decay conditions on $ \phi $ and $ j(r) $ affect the boundary regularity of solutions?

Key findings

• There exists a unique viscosity solution $ u $ to the Dirichlet problem $ Lu = f $ in $ D $, $ u = 0 $ in $ \mathbb{R}^n \setminus D $, where $ D $ is a bounded $ C^{1,1} $ domain.
• The solution satisfies $ u \in C^V(D) $, meaning it is controlled by the renewal function $ V $, which captures the variable-order nature of the kernel.
• The normalized solution $ u/V(d_D) $ belongs to the Hölder space $ C^\alpha(D) $ for some $ \alpha > 0 $, with uniform estimates depending only on $ \|f\|_{L^\infty(D)} $.
• The Hölder seminorm of $ u/V(d_D) $ is bounded by $ C r^{-\beta} / V(r) $ for balls of radius $ r $, where $ r = d_D(x) $, ensuring regularity up to the boundary.
• The proof relies on oscillation estimates in dyadic annuli and interpolation between Hölder and Lipschitz norms of $ 1/V(d_D) $, yielding $ \left[ u/V(d_D) \right]_{C^\beta} \leq C r^{-\beta} / V(r) $.
• The final Hölder continuity of $ u/V(d_D) $ on $ D $ is established by combining local estimates (for small $ |x-y| $) and global oscillation bounds (for large $ |x-y| $), yielding a global $ C^\alpha $ estimate with $ \alpha > 0 $.