$H^1$ Solutions of a class of fourth order nonlinear equations for image processing

Pages: 349 - 366, Volume 10, Issue 1/2, January/February 2004

doi:10.3934/dcds.2004.10.349       Abstract        Full Text (202.0K)              Related Articles

John B. Greer - Department of Mathematics, Duke University, Durham, NC 27708, Department of Mathematics, Univ. of California Los Angeles, Los Angeles, CA 90095, United States (email)
Andrea L. Bertozzi - Department of Mathematics, Duke University, Durham, NC 27708, Department of Mathematics, Univ. of California Los Angeles, Los Angeles, CA 90095, United States (email)

Abstract: Recently fourth order equations of the form $u_t = -\nabla\cdot((\mathcal G(J_\sigma u)) \nabla \Delta u)$ have been proposed for noise reduction and simplification of two dimensional images. The operator $\mathcal G$ is a nonlinear functional involving the 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 problem for $H^1$ initial data.

Keywords:  Fourth order partial differential equations, image processing,anisotropic diffusion, mollifiers, ODEs on a Banach space, higher order nonlinear PDEs.
Mathematics Subject Classification:  35XX

Received: February 2002;      Revised: April 2003;      Published: October 2003.