[Paper Review] Convex integration and infinitely many weak solutions to the Perona-Malik equation in all dimensions

Seonghak Kim, Baisheng Yan|arXiv (Cornell University)|Mar 2, 2015

Numerical methods in inverse problems15 references3 citations

TL;DR

This paper establishes the existence of infinitely many Lipschitz weak solutions to the Perona-Malik equation in all dimensions on smooth bounded convex domains, regardless of whether the initial data is subcritical, supercritical, or transcritical. The authors use a novel convex integration approach combined with a Baire category method to prove that the solution set is residual in a suitable function space, resolving a long-standing open problem in image processing PDEs beyond one dimension.

ABSTRACT

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.

Motivation & Objective

• To resolve the long-standing open problem of existence of weak solutions to the Perona-Malik equation in dimensions greater than one.
• To extend prior results limited to one-dimensional or radially symmetric cases to general smooth bounded convex domains.
• To establish the existence of infinitely many Lipschitz weak solutions for all smooth nonconstant initial data, including transcritical cases.
• To prove a gradient maximum principle for the weak solutions, showing that the gradient remains bounded by a controllable multiple of the initial gradient.

Proposed method

• Reformulate the Perona-Malik equation as a nonhomogeneous partial differential inclusion with a linear constraint and uncontrollable gradient components.
• Apply a Baire category method in a suitable function space to establish the existence of a residual set of solutions.
• Use convex integration techniques to construct approximate solutions by patching oscillatory correctors on small cubes and intervals.
• Establish a density hypothesis via lamination convex hull approximations derived from an explicit formula for a matrix set involved in the PDE inclusion.
• Construct perturbations using piecewise $ C^1 $ functions with controlled time and spatial derivatives to satisfy the weak formulation.
• Verify that the constructed solutions satisfy the weak formulation and approximate the Perona-Malik function $ \sigma(p) = \frac{p}{1+|p|^2} $ in $ L^1 $-norm.

Experimental results

Research questions

• RQ1 Can infinitely many weak solutions exist for the Perona-Malik equation in dimensions greater than one, even when the initial data is transcritical?
• RQ2 Is it possible to construct such solutions using convex integration techniques beyond one-dimensional or radially symmetric settings?
• RQ3 Does the solution set remain dense or residual in a suitable function space under the weak topology, despite the ill-posedness of the original PDE?
• RQ4 Can a gradient maximum principle be preserved for these weak solutions, even when classical solutions do not exist?
• RQ5 What structural properties of the domain (e.g., convexity) are essential for the existence of such solutions?

Key findings

• The initial-Neumann boundary value problem for the Perona-Malik equation admits infinitely many Lipschitz weak solutions on any smooth bounded convex domain in $ \mathbb{R}^n $, for all $ n \geq 1 $.
• The result holds for all smooth nonconstant initial data, including those that are transcritical (i.e., with both subcritical and supercritical regions).
• The weak solutions satisfy the almost gradient maximum principle: $ \|Du\|_{L^\infty(\Omega_T)} \leq \|Du_0\|_{L^\infty(\Omega)} + \lambda $ for any $ \lambda > 0 $, ensuring boundedness of the gradient.
• The solution set is residual in a suitable function space, implying that such solutions are generic in a topological sense.
• The proof relies on a new convex integration scheme that constructs solutions by patching oscillatory correctors on a finite number of small sets, overcoming the lack of gradient control in higher dimensions.
• The key technical advance is the explicit construction of the lamination convex hull of a matrix set, which satisfies the necessary density hypothesis for the Baire category argument.