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

Abstract

Abstract. We prove that for all smooth nonconstant initial data the initialNeumann 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.

On one-dimensional forward–backward diffusion equations with linear convection and reaction

“…In this section, we equip with an important but technical tool for local patching to be used in the proof of the density lemma, Lemma 4.1. The following result is a refinement of the (1 + 1)-dimensional version of a combination of [16,Theorem 2.3 and Lemma 4.5].…”

Convex integration with linear constraints and its applications

“…We begin this section by introducing a pivotal approximation result, Theorem 3.1, for proving the main results of the paper, Theorems 1.1 and 1.3. Its special cases have been successfully applied to some nonstandard evolution problems [15,16,17,14]. Although the proof of Theorem 3.1 already appeared in [14], we include it in Section 6 for the sake of completeness as we make use of the general version of the theorem for the first time in this paper.…”

On weak closure of some diffusion equations

“…Recently, for certain non-monotone functions σ, exact Lipschitz solutions have been constructed for the initial-Neumann problem of (1.1) in [8,9,10]; in such cases, solutions can converge weakly to a function that is not a solution of (1.1).…”

“…In what follows, we denote by g(p, β) the convex hull of function |σ(p)−β| 2 on R n × R n . For p ∈ R n , define 3), which may have some relevance to the existence results in the recent work [8,9,10]. Theorem 1.2.…”