February  2004, 10(1&2): 349-366. doi: 10.3934/dcds.2004.10.349

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

1. 

Department of Mathematics, Duke University, Durham, NC 27708, Department of Mathematics, Univ. of California Los Angeles, Los Angeles, CA 90095, United States, United States

Received  February 2002 Revised  April 2003 Published  October 2003

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.
Citation: John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 349-366. doi: 10.3934/dcds.2004.10.349
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