Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow (Q2905748)

In this paper the authors analyze the following mildly regularized Perona-Malik equation NEWLINE\[NEWLINEu_t=\text{div}\Big ({ \nabla u\over1+|\nabla u|^2}\Big) +\delta\Delta u, (x,t)\in\Omega\times(0,+\infty)NEWLINE\]NEWLINE (where \(\Omega\subseteq \mathbb R^n\) is an open bounded set). The above equation is the gradient-flow equation corresponding to the following family of functionals NEWLINE\[NEWLINEPM_\delta(u):={1\over 2}\int_\Omega \text{log}(1+|\nabla u(x)|^2)\delta x +\delta\int_\Omega |\nabla u(x)|^2 dx,NEWLINE\]NEWLINENEWLINENEWLINE\noindent where \(\delta >0\) is a parameter. The integrand in the above equation``is convex-concave-convex and it grows quadratically at infinity''. The authors prove that the mildly regularized Perona-Malik equation that they introduce ``has a unique global-in-time solution in the sense of Young measures''.NEWLINENEWLINEThe principal scope of the paper is to avoid some problems related to the analytical ill-posedness of the original Perona-Malik equation NEWLINE\[NEWLINEu_t=\text{div}\Big ({ \nabla u\over1+|\nabla u|^2}\Big), (x,t)\in\Omega\times(0,+\infty),NEWLINE\]NEWLINE \noindent which is an example of forward-backward diffusion process well considered in the scientific literature. The authors claim that the obtained analytical results are related to properties put in evidence by numerical experiments given in the references and that the present paper ``provides a rigorous justification of all these properties''; moreover they state ``that the limit of gradient-flow is the gradient-flow of the Gamma-limit'' of the considered sequence of functionals. They prove this ``in the abstract setting of maximal slope curves in metric spaces''. The authors ``believe that the scope of the general result goes far beyond'' the simple application they analyze.