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Article Contents

Young measure solutions of the two-dimensional Perona-Malik equation in image processing

• For a given smooth initial value $u_0$, we construct sequences of approximate solutions $u_j$ in $W^{1,\infty}$ for the well-known Perona-Malik anisotropic diffusion model in image processing defined 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 $X\in\mathbb R^2$. The Perona-Malik diffusion equation is of non-coercive forward-backward type. Our constructed approximate solutions satisfy 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 unit square. We also show, for any non-constant initial value $u_0$ that the approximate solutions $u_j$ do not converge to a solution, rather, they converge weakly to Young measure-valued solutions which can be represented partially explicitly. Our main idea is to convert the equation into a differential inclusion problem.
Mathematics Subject Classification: Primary: 35D05, 68U10; Secondary: 74H20.

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