Abstract
For a given smooth initial value $u_0$, we construct sequences ofapproximate solutions $u_j$ in $W^{1,\infty}$ for the well-knownPerona-Malik anisotropic diffusion model in image processingdefined by $u_t-$ div $ [\rho(|\nabla u|^2)\nabla u]=0$under the homogeneous Neumann condition, where$\rho(|X|^2)X=X/(1+|X|^2)$ for $X\in\mathbb R^2$. ThePerona-Malik diffusion equation is of non-coerciveforward-backward type. Our constructed approximate solutionssatisfy the equation in the sense that $(u_j)_t-$ div$_x [\rho(|\nabla u_j|^2)\nabla u_j]\to 0$ strongly in$W^{-1,p}(Q_T)$ for all $1\leq p<\infty$, where$Q_T=(0,T)\times \Omega$ with $\Omega\subset\mathbb R^2$ the unitsquare. We also show, for any non-constant initial value $u_0$ thatthe approximate solutions $u_j$ do not converge to a solution,rather, they converge weakly to Young measure-valued solutionswhich can be represented partially explicitly. Our main idea is toconvert the equation into a differential inclusion problem.
Published Version
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