In function spaces \(X\) of (equivalence classes) of functions of the type \(f:\Omega\to\mathbb{R}\) with \(\Omega\subset\mathbb{R}^N\) such as Lebesgue spaces \(L^p(\Omega)\) and Sobolev spaces \(W_0^{1,p}(\Omega),W^{1,p}(\Omega)\) we often find sets of the of the type
\begin{equation*}
\mathbf{K}(X)=\{v\in X: |v(x)|\leq \alpha(x) \quad \text{for almost every } x\in\Omega\}
\end{equation*}
where \(\alpha:\Omega\to\mathbb{R}\) is some given function.
Suppose that the boundary of the set \(\Omega\) is smooth enough (at least Lipschitz), then we have the following density results
\begin{align*}
\overline{C^\infty(\Omega)}^{L^p(\Omega)}&=L^p(\Omega), &\overline{C_c^\infty(\Omega)}^{W_0^{1,p}(\Omega)}&=W_0^{1,p}(\Omega), &\text{and} & &\overline{C^\infty(\bar{\Omega})}^{W^{1,p}(\Omega)}&=W^{1,p}(\Omega).
\end{align*}
That is smooth functions on \(\Omega\), with compact support within \(\Omega\), and up to the boundary \(\bar{\Omega}\) are dense in \(L^p(\Omega)\), \(W_0^{1,p}(\Omega)\), and \(W^{1,p}(\Omega)\), respectively. However, in order to obtain the analogous results with contraints as for example
\begin{align*}
\overline{\mathbf{K}(W^{1,p}(\Omega))\cap C^\infty(\bar{\Omega})}^{W^{1,p}(\Omega)}&=\mathbf{K}(W^{1,p}(\Omega)),
\end{align*}
additional requirements on \(\alpha:\Omega\to\mathbb{R}\) are needed.
\(\mathbf{i)}\) There are irregular enough \(\alpha:\Omega\to\mathbb{R}\) that the above fail (see Hintermüller-Rautenberg-Rösel(2017))
\(\mathbf{ii)}\) In general, a dense continuous embedding \(Y\hookrightarrow X\) is not enough to guarantee
\begin{equation*}
\overline{\mathbf{K}\cap Y}^X=\mathbf{K},
\end{equation*}
for \(\mathbf{K}\subset X\).
Regularization of Optimization Problems
Sets of the type \(\mathbf{K}:=\mathbf{K}(X)\) commonly are constraints (on controls or state variables) in optimization problems, i.e., \begin{equation}\tag{$\mathrm{P}$}\label{P} \min_{u\in \mathbf{K}\subset X} J(u). \end{equation} Discretization or regularization approaches to solving \eqref{P} require to consider \begin{equation}\tag{$\mathrm{P}_\gamma$}\label{Pg} \min_{u\in Y\subset X} J(u)+\gamma R(u), \end{equation} where \(\gamma>0\) and \(R(u)=g(\|u\|_Y)\) for some \(g:\mathbb{R}^+_0\to \mathbb{R}^+_0\) strictly increasing. Under natural conditions on \(J\) that provide existence of minimizers, sufficient conditions for the sequence of minimizers \(u_\gamma\) to \eqref{Pg} to converge to a minimizer \(u^*\) of \eqref{P} are given by \begin{equation*} \overline{\mathbf{K}\cap Y}^X=\mathbf{K}. \end{equation*}
Dualization and Automatic Distributed Parameter Selection of Total Generalized Variation via Bilevel Optimization with M. Hintermüller, K. Papafitsoros, and H. Sun
arXiv preprint 2002.05614, 2020.
Variable step mollifiers and applications with M. Hintermüller, and K. Papafitsoros.
arXiv preprint 1910.02751, 2019.
The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow with A. N. Ceretani.
Zeitschrift für angewandte Mathematik und Physik (ZAMP) 70: 14, 2019.
Density of convex intersections and applications with M. Hintermüller, and S. Rösel.
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 473(2205), 2017.
Optimal selection of the regularization function in a weighted total variation model. Part I: Modelling and theory
with M. Hintermüller.
Journal of Mathematical Imaging and Vision, 59(3), 498-514, 2017.
Optimal selection of the regularization function in a weighted total variation model. Part II: Algorithm, its analysis and numerical tests.
with M. Hintermüller, T. Wu and A. Langer.
Journal of Mathematical Imaging and Vision, 59(3), 515-533, 2017.
Analytical aspects of spatially adapted total variation regularisation. with M. Hintermüller, and K. Papafitsoros.
Journal of Mathematical Analysis and Applications, 454(2), pp. 891-935, 2017.
On the Density of Classes of Closed Convex Sets with Pointwise Constraints in Sobolev Spaces with M. Hintermüller.
Journal of Mathematical Analysis and Applications, 426(1), pp. 585-593, 2015.