Evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

Abstract

In this paper we study evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in the framework of metric random walk spaces. This includes as particular cases evolution problems with nonhomogeneous Neumann boundary conditions for the p-Laplacian operator in weighted discrete graphs and for nonlocal operators with nonsingular kernel in R N . A J(x − y)dL N (y) for every Borel set A ⊂ R N , where J : R N → [0, +∞[ is a measurable, nonnegative and radially symmetric function with J = 1. See Section 1.1 for more details.The aim of this paper is to study p-Laplacian type evolution problems like the one given by the following reference model:|u(y) − u(x)| p−2 (u(y) − u(x))dm x (y), x ∈ Ω, 0 < t < T,(1.1)

Gradient flows in metric random walk spaces

“…The following integration by parts formula follows by the reversibility of ν with respect to m. Proposition 5.2 [47] Let j ∈ {1, 2}. Let u be a ν-measurable function such that…”

“…Specifically, together with the existence and uniqueness of solutions to the aforementioned problems, a wide variety of their properties have been studied (some of which are listed in the contents section), as well as the nonlocal diffusions operators involved in them. This survey is mainly based on the results that we have obtained in [46,47] (see also the more recent work [55]) and [48]. Related to the above problems, we have also studied (see [49]) the (BV , L p )-decomposition, p = 1 and p = 2, of functions in metric random walk spaces.…”

“…Let ϕ ∈ L m,∞ (∂ m , ν).The operator A m a p ,ϕ is completely accretive and satisfies the range condition L p ( , ν) ⊂ R(I + A m a p ,ϕ ). Theorem 5.11[47] Let ϕ ∈ L m,∞ (∂ m , ν). Then, we have L ∞ ( , ν) ⊂ D(A m a p ,ϕ ) and, consequently, D(A m a p ,ϕ ) L p ( ,ν) = L p ( , ν).…”