Given a function $$f(x, y): \mathbb{R} \left[kg \right] \times \mathbb{R} \left[K \right] \mapsto \mathbb{R} \left[m \right]$$ where the units of the variables $x, y$ and of the function $f(x, y)$ are written in [square brackets], what are the units of the Laplacian $\Delta f(x,y)$? If the Laplacian of the function $f(x,y)$ is to be computed as
$$ \Delta f(x,y) = \frac{\partial^2 f}{\partial x^2}(x,y) + \frac{\partial^2 f}{\partial y^2}(x,y) $$
then the units of the Laplacian would be $m\cdot kg^{-2} + m\cdot K^{-2}$, which does not make sense to me from the perspective of dimensional analysis, since you cannot add two numbers which represent different physical quantities into a single number.
Is Laplacian defined only for such functions of multiple variables, where each variable represents the same physical quantity (and of course for functions of single variable)?
Edit:
Given the comments below, computing the Laplacian does indeed make "physical" sense only if the domain of the function consists of variables of the same physical units.
However, does there exist a practical (non-toy) example where we compute the Laplacian of a function despite the physical units of the input variables differ and the result has real-world usage?
