Nonlocal Operators

Abstract

Following Gilboa–Osher [50] (see also [14]), we introduce the following nonlocal operators. For a function $u : \mathbb {R}^N \rightarrow \mathbb {R}$, we define its nonlocal gradient as the function $\nabla _J u : \mathbb {R}^N \times \mathbb {R}^N \rightarrow \mathbb {R}$ defined by:

$$\displaystyle (\nabla _Ju) (x,y) = J(x-y) (u(y) - u(x)), \qquad x, y \in \mathbb {R}^N. $$

And for a function $\mathbf {z} : \mathbb {R}^N \times \mathbb {R}^N \rightarrow \mathbb {R}$, its nonlocal divergence ${\mathrm {div}}_J \mathbf {z} : \mathbb {R}^N \rightarrow \mathbb {R}$ is defined as:

$$\displaystyle ({\mathrm {div}}_J \mathbf {z})(x) = \frac {1}{2} \int _{\mathbb {R}^N} (\mathbf {z}(x,y) - \mathbf {z}(y,x)) J(x-y) dy. $$