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Nonlocal Operators

  • José M. Mazón
  • Julio Daniel Rossi
  • J. Julián Toledo
Chapter
Part of the Frontiers in Mathematics book series (FM)

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

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • José M. Mazón
    • 1
  • Julio Daniel Rossi
    • 2
  • J. Julián Toledo
    • 3
  1. 1.Departamento de Análisis MatemáticoUniversitat de ValènciaValenciaSpain
  2. 2.Departamento de MatemáticasUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Departamento de Análisis MatemáticoUniversitat de ValènciaValènciaSpain

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