Considering a scalar field in a plane (pressure vs. location) $P({\rm r})$ where ${\rm r}=(x,y)$ then the following surface integral gives the surface deformation due to the pressure in the elastic half space
$$ \delta = - C \iint_{A} \frac{P({\rm r})}{\|{\rm r}\|}\,{\rm d}A $$
Is there a way to reverse this equation into a differential form. Something like
$$ \nabla^2 \delta = \ldots C\, P(\vec{r})$$
where $\nabla^2 = \ldots \frac{\partial^2 }{\partial x^2} \ldots \frac{\partial^2 }{\partial x \partial y} \ldots \frac{\partial^2 }{\partial y^2} $
In the situations I am looking at I know the shape of $\delta$ and need to find the shape of $P({\rm r})$. Numerically this is very intensive because the discretization of the domain of $\rm r$ results in dense matrix with several thousand elements than need inversion. In a differential form I could try a different approach to solving this problem (using finite differences for example).
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