Elliptic equation with lower dimensional data

Question

I'm looking at $u - \Delta^2 u = f$ with homogeneous boundary and Neumann conditions on the unit square, $\Omega$. In particular, I'm looking at the case where $f\in L^2(S)$ is only supported on a closed 1-dimensional curve $S\subset \Omega$. I'm struggling to formulate a weak version, since $f$ has 0 Lebesgue measure with respect to $\Omega$. Is there a way of writing down a weak version of a PDE problem when the data is defined only with respect to a lower-dimensional Lebesgue measure?

Answer 1

I think the reason this hasn't been answered is that you haven't really specified what $f$ is as a distribution (i.e. what you want it to do to test functions); the answer depends on whether you're thinking of $f$ as a density with respect to Lebesgue measure on $\Omega$ (in which case you get the same weak formulation as the homogeneous equation), or as a density with respect to the pushforward measure given by

$$ \mu(E) = \sigma(i_S^{-1}E) $$

where $i_S: S\to \Omega$ is the inclusion map and $\sigma$ is Lebesgue measure on $S$. In this second case, the weak formulation is the same as usual except that the integral involving $f$ is with respect to $\mu$ rather than Lebesgue measure on $\Omega$.