[Paper Review] Obstacle problems for nonlocal operators with singular kernels

Xavier Ros‐Oton, Marvin Weidner|arXiv (Cornell University)|Aug 3, 2023

Nonlinear Partial Differential Equations4 citations

TL;DR

This paper establishes optimal $C^{1,s}$ regularity for solutions and $C^{1, heta}$ regularity of the free boundary in nonlocal obstacle problems governed by integro-differential operators with singular, non-comparable kernels—extending classical results beyond the fractional Laplacian framework by developing new nonlocal tools that bypass the need for Harnack inequalities or extension problems.

ABSTRACT

In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in contrast to all previous known results, we are able to treat nonlocal operators whose kernels are not necessarily pointwise comparable to the one of the fractional Laplacian. Such operators might be very anisotropic in the sense that they "do not see" certain directions at all, or might have substantial oscillatory behavior, causing the nonlocal Harnack inequality to fail.

Motivation & Objective

• To extend the regularity theory of nonlocal obstacle problems beyond the fractional Laplacian, where kernels are not pointwise comparable to the fractional Laplacian’s kernel.
• To address the lack of regularity results for nonlocal operators with singular, anisotropic, or oscillatory kernels that violate the standard Harnack inequality.
• To establish optimal $C^{1,s}$ regularity of solutions and $C^{1, heta}$ regularity of the free boundary in the absence of the pointwise comparability condition $K_{ ext{asymp}}$.
• To develop new analytical tools that do not rely on the extension problem or monotonicity formulas, applicable to general stable operators with singular kernels.
• To resolve the regularity of free boundaries for nonlocal operators with kernels that may vanish in certain directions or exhibit $L^p$-singularities on the sphere $\mathbb{S}^{n-1}$.

Proposed method

• Introduces a new method to analyze the regularity of solutions to obstacle problems $\min\{Lu, u - \phi\} = 0$ where $L$ is a stable integro-differential operator with homogeneous kernel $K(y) = |y|^{-n-2s}K(y/|y|)$.
• Establishes a priori $C^{1,s}$ regularity estimates for solutions via a blow-up and compactness argument, relying on the structure of the kernel and the obstacle's smoothness.
• Uses a nonlocal version of the Nirenberg–Calderón type approximation to control the behavior of the gradient ratio $\partial_i u / \partial_n u$ near the free boundary.
• Applies a new nonlocal decay estimate for the difference $[\partial_i u - K \partial_n u](x)$, which controls the oscillation of the normal vector to level sets.
• Combines gradient estimates with a nonlocal version of the boundary Harnack principle to derive Hölder regularity of the normal vector, implying $C^{1, heta}$ free boundary regularity.
• Relies on a scaling argument and compactness to show that solutions converge to a homogeneous blow-up limit of the form $(x \cdot e)_+^{1+s}$, confirming the free boundary's regularity at such points.

Experimental results

Research questions

• RQ1 Can optimal $C^{1,s}$ regularity be established for solutions to nonlocal obstacle problems when the kernel $K$ is not pointwise comparable to the fractional Laplacian kernel?
• RQ2 What regularity can be achieved for the free boundary $\partial\{u > \phi\}$ when the nonlocal operator $L$ is anisotropic or has singularities on $\mathbb{S}^{n-1}$?
• RQ3 Is it possible to prove $C^{1, heta}$ regularity of the free boundary without relying on the extension problem or monotonicity formulas?
• RQ4 How can one control the oscillation of the normal vector to level sets in the absence of the nonlocal Harnack inequality?
• RQ5 Can the regularity theory be extended to operators with kernels that vanish in certain directions or have $L^p$-singularities on the sphere?

Key findings

• The paper establishes optimal $C^{1,s}$ regularity for solutions $u$ to the obstacle problem $\min\{Lu, u - \phi\} = 0$ even when the kernel $K$ violates the pointwise comparability condition $K_{\asymp}$.
• It proves that the free boundary $\partial\{u > \phi\}$ is $C^{1,\theta}$ regular near regular points, under the same general kernel assumptions.
• The authors derive a new nonlocal decay estimate for the gradient ratio $\partial_i u / \partial_n u$, which controls the Hölder seminorm of the normal vector to level sets.
• The method avoids the use of the extension problem and monotonicity formulas, enabling the analysis of operators with kernels that may vanish on subsets of $\mathbb{S}^{n-1}$ or be singular.
• The result applies to operators such as $L_1$ (double-cone supported), $L_2$ (with $L^p$-singular kernel), and $L_3$ (anisotropic fractional Laplacian), which were previously outside the scope of regularity theory.
• The proof relies on a new compactness argument and scaling, showing that blow-up limits of solutions converge to homogeneous solutions of the form $(x \cdot e)_+^{1+s}$, confirming the expected asymptotic behavior at regular points.