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Article Contents

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

• 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.
Mathematics Subject Classification: 35XX.

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