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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.

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3 Answers 3

up vote 5 down vote accepted

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.

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.

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The place to start is: Hormander, Analysis of Linear partial differential operators, vols. I-II (if your coefficients are constant), and further volumes for non-constant coefficients.

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Almost any book on pseudodifferential operators will discuss how to prove existence, uniqueness, and regularity for the elliptic PDE. I'm not sure about the parabolic PDE.

I learned this stuff from "Introduction to the Theory of Linear Partial Differential Equations" by Chazarain and Piriou.

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