# Regularization in Sobolev Spaces with Fractional Order

Ute Assmann* and Arnd Rösch

We study the minimization of a quadratic functional where the Tichonov regularization term is given in the $H^s$-norm with a fractional parameter $s>0$. Moreover, pointwise bounds for the unknown solution are given. We introduce a multilevel approach as an equivalent norm concept, that is usefull on the one hand for showing higher regularity of the solution of the variational inequality. This regularity implies the existence of regular Lagrange multipliers in function space. On the other hand the multiplier approach of the $H^s$-norm is suitable for a numerical treatment of the problem, which is the topic of the last part of the talk.

Mathematics Subject Classification: 49N60 49N15 49K20 49N45

Keywords: Optimal control, variational inequalities, multilevel operator, Lagrange multipliers, parameter identification, inverse problems

Minisymposion: Stability, Sensitivity and Error Analysis for Optimal Control Problems