$H^1$ Solutions of a class of fourth order nonlinear equations for image processing
John B. Greer, Andrea Louise Bertozzi · Discrete and Continuous Dynamical Systems · 2004
Recently fourth order equations of the form$u_t = - abla\cdot((\mathcal G(J_\sigma u)) abla \Delta u)$ have been proposedfor noise reduction and simplification of two dimensional images.The operator $\mathcal G$ is a nonlinear functional involvingthe gradient or Hessian of its argument, with decay in the far field.The operator $J_\sigma$ is a standard mollifier.Using ODE methods on Sobolev spaces,we prove existence and uniqueness of solutions of this problemfor $H^1$ initial data.