[Paper Review] Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications

Anup Biswas, Mitesh Modasiya|arXiv (Cornell University)|Oct 13, 2021

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper introduces a general class of mixed local-nonlocal elliptic operators combining the Laplacian with a general nonlocal jump-type operator, establishing existence, uniqueness, and maximum principles for Dirichlet problems. It proves a Faber-Krahn-type inequality and a one-dimensional symmetry result (generalizing Gibbons' conjecture), significantly extending prior work on operators like $-\Delta + (-\Delta)^s$ to broader Lévy jump measures.

ABSTRACT

In this article we consider a class of non-degenerate elliptic operators obtained by superpositioning the Laplacian and a general nonlocal operator. We study the existence-uniqueness results for Dirichlet boundary value problems, maximum principles and generalized eigenvalue problems. As applications to these results, we obtain Faber-Krahn inequality and a one-dimensional symmetry result related to the Gibbons' conjecture. The latter results substantially extend the recent results of Biagi et.\ al. [7,9] who consider the operators of the form $-Δ+ (-Δ)^s$ with $s\in (0, 1)$.

Motivation & Objective

• To develop a general framework for mixed-order elliptic operators combining local (Laplacian) and nonlocal (general jump-type) components.
• To establish existence and uniqueness of viscosity solutions for Dirichlet problems with such operators.
• To derive maximum principles and generalized eigenvalue problems for the new class of operators.
• To extend the Faber-Krahn inequality and one-dimensional symmetry results (Gibbons' conjecture) beyond the fractional Laplacian case.
• To generalize prior results on $-\Delta + (-\Delta)^s$ to operators with arbitrary Lévy jump kernels satisfying integrability conditions.

Proposed method

• The operator is defined as $Lu = \Delta u + \int_{\mathbb{R}^d} (u(x+y) - u(x) - \mathbf{1}_{\{|y|\leq 1\}} y\cdot\nabla u(x)) j(y) \, dy$, with $j$ a general jump kernel satisfying $\int (1\wedge |y|^2)j(y)\,dy < \infty$.
• The analysis relies on the connection to Lévy processes: $X = B + Y$, where $B$ is Brownian motion and $Y$ is a pure-jump Lévy process with jump measure $j(y)\,dy$, leading to a generator of the form (1.1).
• Viscosity solutions are used throughout, defined via comparison with smooth test functions achieving strict minima or maxima.
• The Feynman-Kac formula is employed to represent the solution as $u(x) = \mathbb{E}_x[\int_0^{\uptau} f(X_t)\,dt] + \mathbb{E}_x[g(X_{\uptau})]$, with $\uptau$ the first exit time from the domain.
• The moving plane method is applied to prove one-dimensional symmetry results, extending Gibbons' conjecture to general nonlocal operators.
• A priori estimates and convergence arguments for approximating sequences of processes are used to prove continuity and convergence of solutions.

Experimental results

Research questions

• RQ1 Can existence and uniqueness of viscosity solutions be established for mixed local-nonlocal elliptic operators with general jump kernels?
• RQ2 Do maximum principles hold for this class of operators under mild integrability conditions on the jump kernel?
• RQ3 Can the Faber-Krahn inequality be extended to operators combining Laplacian and general nonlocal terms?
• RQ4 Does the one-dimensional symmetry result (Gibbons' conjecture) hold for this broader class of operators beyond the fractional Laplacian?
• RQ5 How do the principal eigenvalue and spectral properties behave for such mixed-order operators?

Key findings

• The paper establishes existence and uniqueness of viscosity solutions to the Dirichlet problem $Lu = -f$ in $D$, $u = g$ on $D^c$, under assumption (A1) and regularity of $f$ and $g$.
• A Feynman-Kac-type representation formula is derived: $u(x) = \mathbb{E}_x[\int_0^{\uptau} f(X_t)\,dt] + \mathbb{E}_x[g(X_{\uptau})]$, linking the solution to the first exit distribution of the Lévy process.
• A generalized maximum principle is proven for the mixed operator, ensuring that non-positive solutions achieve their maximum on the boundary.
• The principal eigenvalue $\lambda_1$ exists and is positive, with a corresponding positive eigenfunction, under appropriate conditions.
• A Faber-Krahn-type inequality is established, showing that the first eigenvalue is minimized for balls among domains of equal volume.
• A one-dimensional symmetry result is proven: under suitable conditions, solutions to semilinear equations $Lu = f(u)$ are symmetric in the direction of the first coordinate, extending Gibbons' conjecture to general nonlocal operators.