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A unified view of nonlinear, nonlocal operators and qualitative properties of associated elliptic and parabolic problems

Ralph Chill, Mahamadi Warma

TL;DR

This work develops a unified, variational framework for nonlinear nonlocal elliptic and parabolic problems by realizing operators as $j$-subgradients of convex energies on Musielak–Orlicz–Sobolev spaces. It establishes well-posedness of the abstract gradient system, develops a nonlinear Dirichlet-form theory, and derives qualitative properties including comparison principles, domination, and ultracontractivity, while incorporating Robin and Dirichlet exterior conditions. The theory is demonstrated through diverse nonlocal operators—fractional, regional, graph-based, and metric random walk spaces—covering Euclidean domains, graphs, and beyond, with explicit energy forms and boundary/exterior representations. These results provide a broad, flexible toolkit for analyzing nonlinear nonlocal diffusion, enabling new operator classes and rigorous analysis across multiple settings and geometries.

Abstract

We put together a general framework to deal with elliptic and parabolic equations associated with (nonlinear) nonlocal (fractional order) operators. Many well-known nonlocal operators enter into our framework, and in addition one may introduce many other, new nonlocal operators that have not yet been considered in the literature. We use the abstract theory of (nonlinear) semigroups generated by subgradients of proper, lower semicontinuous and convex functionals on Hilbert spaces to build a rigorous and applicable framework that works for many classical elliptic operators but also nonlocal or sometimes fractional operators. After recalling the notion of a nonlinear semigroup generated by subgradients and $j$-subgradients of the associated energy functions, we introduce a general class of (nonlinear) nonlocal elliptic type operators and define rigorously subgradients and $j$-subgradients of such functionals that generate (nonlinear) submarkovian semigroups and hence, the abstract Cauchy problem associated with these subgradients and/or $j$-subgradients is wellposed. The existence and the qualitative properties of solutions to these Cauchy problems and the corresponding semigroups are investigated. More precisely, we show some comparison and maximum principles, submarkovian, domination, ultracontractivity properties, and some Hölder type estimates for these semigroups of operators. These results are usually useful in several branches of pure and applied partial differential equations. We finish the paper by giving several examples of nonlocal operators in Euclidean spaces, graphs, metric random walk spaces, fractional Brownian motions, and Lévy flights, that fit in our general framework.

A unified view of nonlinear, nonlocal operators and qualitative properties of associated elliptic and parabolic problems

TL;DR

This work develops a unified, variational framework for nonlinear nonlocal elliptic and parabolic problems by realizing operators as -subgradients of convex energies on Musielak–Orlicz–Sobolev spaces. It establishes well-posedness of the abstract gradient system, develops a nonlinear Dirichlet-form theory, and derives qualitative properties including comparison principles, domination, and ultracontractivity, while incorporating Robin and Dirichlet exterior conditions. The theory is demonstrated through diverse nonlocal operators—fractional, regional, graph-based, and metric random walk spaces—covering Euclidean domains, graphs, and beyond, with explicit energy forms and boundary/exterior representations. These results provide a broad, flexible toolkit for analyzing nonlinear nonlocal diffusion, enabling new operator classes and rigorous analysis across multiple settings and geometries.

Abstract

We put together a general framework to deal with elliptic and parabolic equations associated with (nonlinear) nonlocal (fractional order) operators. Many well-known nonlocal operators enter into our framework, and in addition one may introduce many other, new nonlocal operators that have not yet been considered in the literature. We use the abstract theory of (nonlinear) semigroups generated by subgradients of proper, lower semicontinuous and convex functionals on Hilbert spaces to build a rigorous and applicable framework that works for many classical elliptic operators but also nonlocal or sometimes fractional operators. After recalling the notion of a nonlinear semigroup generated by subgradients and -subgradients of the associated energy functions, we introduce a general class of (nonlinear) nonlocal elliptic type operators and define rigorously subgradients and -subgradients of such functionals that generate (nonlinear) submarkovian semigroups and hence, the abstract Cauchy problem associated with these subgradients and/or -subgradients is wellposed. The existence and the qualitative properties of solutions to these Cauchy problems and the corresponding semigroups are investigated. More precisely, we show some comparison and maximum principles, submarkovian, domination, ultracontractivity properties, and some Hölder type estimates for these semigroups of operators. These results are usually useful in several branches of pure and applied partial differential equations. We finish the paper by giving several examples of nonlocal operators in Euclidean spaces, graphs, metric random walk spaces, fractional Brownian motions, and Lévy flights, that fit in our general framework.
Paper Structure (15 sections, 33 theorems, 279 equations)

This paper contains 15 sections, 33 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Semigroups generated by negative subgradients
  3. Realization of nonlocal elliptic operators as subgradients on $L^2 (\Omega )$
  4. Comparison principles, maximum principles and nonlinear Dirichlet forms
  5. Interlude: Regular Dirichlet forms and the capacity induced by the space $W^{\Phi ,2} (\hat{\Omega} , \Omega )$
  6. Potentials, Robin exterior conditions and Robin boundary conditions
  7. Domination
  8. Ultracontractivity for Dirichlet and Neumann exterior conditions
  9. Dirichlet exterior and boundary conditions
  10. Neumann exterior conditions
  11. Examples and Remarks
  12. Fractional powers of linear elliptic operators in domains of $\mathbb R^N$
  13. Other nonlocal operators in domains of $\mathbb R^N$
  14. Nonlocal operators on graphs
  15. Nonlocal operators on metric random walk spaces

Key Result

Lemma 3.1

Let $\Phi :\hat{\Omega} \times\hat{\Omega}\times\mathbb{R} \to [0,\infty]$ be a function satisfying the measurability and convexity condition cond.phi, and the $\Delta_2$-condition. Suppose that the set ${\mathbf S}_\Phi$ satisfies the thickness condition, that is, Then the space $W^{\Phi ,2} (\hat{\Omega} , \Omega )$ is a Banach space. Moreover, every convergent sequence $(\hat{u}_k)$ in $W^{\P

Theorems & Definitions (85)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 75 more