Abstract
Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each xin X, which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.
Highlights
The digital world has brought with it many different kinds of data of increasing size and complexity
Following the implementation of the graph Laplacian in the development of spectral clustering in the seventies, the theory of partial differential equations on graphs has obtained important results in this field. This has prompted a big surge in the research of nonlocal partial differential equations
Interest has been further bolstered by the study of problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes among other problems
Summary
The digital world has brought with it many different kinds of data of increasing size and complexity. Nonlocal diffusion problems of p-Laplacian type with homogeneous Neumann boundary conditions have been studied in nonlocal models in RN associated to a non-singular kernel (see, for example, [4,5]) and in weighted discrete graphs (see, for example, the work of Hafiene et al [35]) with the following formulation: ut (t, x) = |u(y) − u(x)|p−2(u(y) − u(x))dmx (y), x ∈ , 0 < t < T . In [34], Gunzburger and Lehoucq develop a nonlocal vector calculus with applications to linear nonlocal problems in which the nonlocal Neumann boundary condition considered can be rewritten as (u(y) − u(x))dmx (y) = φ(x), x ∈ ∂m Another interesting approach is proposed by Dipierro et al in [25] for the particular case of the fractional Laplacian diffusion ( the idea can be used for other kernels) with the following Neumann boundary condition, that we rewrite in the context of metric random walk spaces,. In our work we study Problem (1.3) with the nonhomogeneous Neumann boundary conditions of Gunzburger–Lehoucq type and of Dipierro–Ros-Oton–Valdinoci type
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