UTMS 2006 – 5 March 29 , 2006 Regularity of the Tikhonov regularized solutions and the exact solution

We discuss an operator equation Kf = g where X, Y are reflexive Banach spaces of functions in bounded domains Ω ⊂ RN and Ω0, and K : X −→ Y has no continuous inverse. Let V and V1 be another reflexive Banach spaces of functions in Ω and D ⊂ Ω respectively such that the embedding V ⊂ X is continuous. In order to stably reconstruct g by noise data gδ with ‖g − gδ‖Y ≤ δ: noise level, we consider the Tikhonov regularization: Minimize ‖Kf − gδ‖Y + α‖f‖V . We prove that if α = c0δ with a constant c0 > 0 and the V1-norms of the regularized solutions fδ are bounded uniformly in δ, then the exact solution f is in V1. This property can be applied to the determination of non-smooth points of a function f for example in the case of X = L2(Ω), V = H`(Ω) and V1 = H`(D) with ` ∈ N and small ball D. §