Convex Integration and Infinitely Many Weak Solutions to the Perona-Malik Equation in All Dimensions

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 but some uncontrollable components of gradient. We establish a general existence result by a suitable Baire 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.