Regularization in Sobolev Spaces with Fractional Order

We study the minimization of a quadratic functional where the Tichonov regularization term is an H ( s )-norm with a fractional s > 0. Moreover, pointwise bounds for the unknown solution are given. A multilevel approach as an equivalent norm concept is introduced. We show higher regularity of the solution of the variational inequality. This regularity is used to show the existence of regular Lagrange multipliers in function space. The theory is illustrated by two applications: a Dirichlet boundary control problem and a parameter identification problem.