If I want to learn about existence of weak solutions to PDEs of the form
$$u\_t + Au = f$$
or
$$Au = f$$
where $A$ is elliptic and $f$ is a measure, where do I start? I know the Galerkin method for parabolic PDEs with right hand side in $L^2(0,T;V')$ or $L^2(0,T;H)$ (where $V \subset H$). But now I want to a step deeper into PDE theory and consider measures instead.


In the literature I have come across
1) $f$ = Radon measure
2) $f$ = Young measure
...


and I don't really what how to begin. Can anyone recommend me a text/source to learn this kind of stuff, or give me a quick overview about this area? I am particularly interested in $f = \text{Dirac delta}$ but maybe that can come later. I would appreciate a book where as little measure theory as required is there since it is not my strong point. Thank you.



PDEs involving measures; where to begin?
Weak solutions for PDEs with Radon measures as right-hand sides can be obtained by a duality technique, which roughly proceeds as follows. Assume that the adjoint operator $A^\*$ acts as isomorphism from $W^{1,q}\_0(\Omega)$ to $W^{-1,q}(\Omega)$ for some $q>n$ with $\Omega\subset\mathbb{R}^n$. (This is the case if the coefficients of $A$ and the domain $\Omega$ is sufficiently smooth.) The closed range theorem and reflexivity of the spaces then implies that $A$ is an isomorphism from $W^{1,q'}\_0(\Omega)$ to $W^{-1,q'}(\Omega)$ for $1/q'+1/q = 1$, i.e., $q'< n/(n-1)$. Since $W^{1,q}\_0(\Omega)$ is continuously and densely embedded into $C\_0(\Omega)$ for $q>n$, we have the dual embedding of $C\_0(\Omega)^\*$ (which can be identified by the Riesz theorem with the space of Radon measures) into $W^{-1,q'}(\Omega)$. Together, this yields for every Radon measure $f\in C\_0(\Omega)^\*\hookrightarrow W^{-1,q'}(\Omega)$ a unique solution $u\in W^{1,q'}(\Omega)$ to $Au=f$.


If $A^\*$ lacks this maximal regularity, the solution is no longer unique, and you need to either extend the space of test functions (leading to *solutions in the sense of Stampacchia*) or restrict the space of solutions (leading to *solutions in the sense of Boccardo–Gallouët*); both approaches are equivalent. All this is very nicely explained in


[Meyer, C., Panizzi, L., and Schiela, A. (2011). Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control. Numerical Functional Analysis and Optimization 32.9, pp. 983–1007.](http://dx.doi.org/10.1080/01630563.2011.587074)


For parabolic and nonlinear equations, see, e.g.,


[Boccardo, L., and Gallouët, T. (1989). Non-linear elliptic and parabolic equations involving measure data. Journal of Functional Analysis 87, pp. 149–169.](http://dx.doi.org/10.1016/0022-1236%2889%2990005-0)


