  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.
