The operator
$$ \Delta:=\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}+\frac{\partial^2 }{\partial z^2} $$is called the Laplacian operator. It appears in Laplace's equation and in Poisson's equation. Also appears in the heat equation.
It is usually written as $\Delta=\nabla^2=\nabla \cdot \nabla$ . Also observe that $\nabla \cdot$ is the divergence in the 3D case.
In the 2D case, I understand it like a measure of the "force" existing in every point of a kind of "membrane" due to the influence of the surrounding points
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Author of the notes: Antonio J. Pan-Collantes
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