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

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.