ETD

• calculus
• double well
• functional
• integral
• Malik
• minimization
• parameter
• Perona
• variations
• Young

We investigate the shape of minimum values of a Perona-Malik functional depending on a positive parameter under Dirichlet boundary conditions; we search for the asymptotic behaviour of the sequence of minimum values with respect to the parameter when the parameter takes small values. To this aim we discuss minimization problem associated with an auxiliary functional obtained by changing the first derivative term of the Lagrangian; in this way we obtain the well know double well functional. We deal with the search of a minimizing sequence for such a functional deprived of the scale parameter, getting to introduce Young measures as a tool to investigate ist limiting behaviour; then at first we face the complete problem by a qualitative point of view and then we try to formalize the results we get. At this point we search for a generalization holding for the Perona-Malik functional, also exploiting some Gamma-convergence results.