[Paper Review] Local regularity for nonlocal equations with variable exponents

Jamil Chaker, Minhyun Kim|arXiv (Cornell University)|Jul 13, 2021

Nonlinear Partial Differential Equations44 references4 citations

TL;DR

This paper establishes local boundedness and Hölder regularity for weak solutions to nonlocal equations with variable exponents $p(x,y)$, proving that solutions are locally bounded when $p$ is continuous and locally H"older continuous under a log-H"older-type condition on $p$ inside the domain and an additional condition on $p$ outside the domain. The results extend classical regularity theory to nonlocal, variable-exponent settings using De Giorgi-type iteration and Caccioppoli estimates.

ABSTRACT

In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler--Lagrange equations. We show that weak solutions are locally bounded when the variable exponent $p$ is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on $p$. On the one hand, the class of admissible exponents is assumed to satisfy a log-Hölder-type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on $p$ outside the domain.

Motivation & Objective

• To establish local regularity theory for minimizers of nonlocal variational functionals with variable exponents $p(x,y)$.
• To extend classical regularity results from local $p(x)$-Laplacian equations to nonlocal fractional $p(x,y)$-Laplacian equations.
• To identify minimal conditions on $p(x,y)$ ensuring boundedness and H"older continuity of weak solutions.
• To address the challenge of nonlocality by introducing an exterior condition on $p$ beyond the domain.

Proposed method

• Introduces a nonlocal variational functional with variable exponent $p(x,y)$, modeling fractional $p(x,y)$-Laplacian equations.
• Defines weak solutions to the Euler–Lagrange equation associated with the functional and establishes their connection to minimizers.
• Applies a Caccioppoli-type estimate to control oscillation of solutions in balls, enabling iterative regularity improvement.
• Employs a De Giorgi-type iteration scheme to prove Hölder continuity, relying on decay of superlevel sets.
• Imposes two key conditions on $p$: (P1) a log-H"older-type condition on $p$ within the domain, and (P2) a monotonicity condition on $p$ outside the domain.
• Uses dyadic iteration and testing with truncated functions to control the growth of superlevel sets and derive Hölder estimates.

Experimental results

Research questions

• RQ1 Under what conditions on the variable exponent $p(x,y)$ are weak solutions to nonlocal $p(x,y)$-Laplacian equations locally bounded?
• RQ2 Can the classical log-H"older continuity condition for local $p(x)$-Laplacians be extended to nonlocal, variable-exponent problems?
• RQ3 What additional conditions on $p(x,y)$ are required to ensure Hölder regularity in the nonlocal setting beyond the interior log-H"older condition?
• RQ4 How does the behavior of $p(x,y)$ outside the domain affect the regularity of solutions in nonlocal problems?
• RQ5 Is the De Giorgi iteration method applicable to nonlocal equations with variable exponents, and what modifications are needed?

Key findings

• Weak solutions to the nonlocal $p(x,y)$-Laplacian equation are locally bounded whenever $p(x,y)$ is continuous and bounded away from 1 and infinity.
• Local Hölder continuity of bounded weak solutions is established under the log-H"older-type condition (P1) on $p$ within the domain.
• An additional condition (P2) on the behavior of $p(x,y)$ outside the domain is necessary to control nonlocal interactions and ensure regularity.
• The proof relies on a refined Caccioppoli estimate and a modified De Giorgi iteration scheme that accounts for variable exponents and nonlocality.
• The log-H"older-type condition (P1) is shown to be sharp in the sense that failure leads to loss of Hölder regularity, mirroring the local case.
• The results generalize classical regularity theory for local $p(x)$-Laplacians to the nonlocal, variable-exponent setting, providing a complete local regularity framework.