$H^1$ Solutions of a class of fourth order nonlinear equations for image processing
$H^1$ Solutions of a class of fourth order nonlinear equations for image processing
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DOI：
10.3934/dcds.2004.10.349
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发表时间：
2003-10
影响因子：
1.1
通讯作者：
J. Greer;A. Bertozzi
中科院分区：
文献类型：
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作者：
J. Greer;A. Bertozzi
最近提出了四阶方程u_t = -\nabla\cdot（（\mathcal G（J_\sigma u））\nabla \Delta u）$用于二维图像的降噪和简化。算子$\mathcal G$是一个非线性泛函，涉及其自变量的梯度或海森，在远场衰减。运算符$J_\sigma$是一个标准的软化器。利用Sobolev空间上的ODE方法，我们证明了该问题对于H^1 $初值的解的存在唯一性。
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.