高维Perona-Malik方程的凸积分与无穷多个弱解
Convex integration and infinitely many weak solutions to the Perona-Malik equation in all dimensions
摘要：
本文证明了图像处理中Perona-Malik方程的初Neumann边值问题在光滑有界凸区域上，对所有光滑的非常数初值，具有无穷多个Lipschitz弱解.除了一维问题外，这样的存在性结果还不为人所知。我们的方法是由Perona-Malik方程转化为一个非齐次部分微分包含与线性约束和不可控的梯度分量。在枢轴密度假设下，利用适当的Baire范畴方法，得到了一个普遍存在的结果.最后，我们通过凸积分来实现这个密度假设，该凸积分是基于某些矩阵集的层凸包的显式公式的近似。
英文摘要：
We prove that for all smooth nonconstant initial data the initial-Neumann boundary value problem for the Perona-Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona-Malik equation as a nonhomogeneous partial differential inclusion with linear constraint and uncontrollable components of gradient. We establish a general existence result by a suitable Baire's category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved.
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