Abstract: We address density properties of intersections of convex sets in Sobolev spaces. The constraint sets are determined by pointwise bounds on function values, their gradient or divergence, and they are commonly found in infinite dimensional control and optimization problems, and variational inequalities. Using Gamma-convergence we show that these density issues arise (as necessary and sufficient conditions) for consistency of regularization and discretization, and in dualization of these type of problems. It is shown that a dense embedding is not enough to guarantee density, and we provide counterexamples for discontinuous obstacles and minimum regularity results for obstacles in Sobolev spaces. Additionally, we consider concrete applications on image reconstruction, and elastoplasticity.
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