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The purpose of this work is to illustrate how the theory of Muckenhouptweights, Muckenhoupt weighted Sobolev spaces and the corresponding weightednorm inequalities can be used in the analysis and discretization of PDEconstrained optimization problems. We consider: a linear quadratic constrainedoptimization problem where the state solves a nonuniformly elliptic equation; aproblem where the cost involves pointwise observations of the state and onewhere the state has singular sources, e.g. point masses. For all three exampleswe propose and analyze numerical schemes and provide error estimates in two andthree dimensions. While some of these problems might have been consideredbefore in the literature, our approach allows for a simpler, Hilbertspace-based, analysis and discretization and further generalizations.

Some applications of weighted norm inequalities to the error analysis of PDE constrained optimization problems