On constrained equations with singular diffusivity
Let \(E(u)\) denote the total variation of a mapping \(u:\Omega\subset\mathbb R^N\rightarrow\mathbb R^N\), that is, \(E(u)=\int_\Omega | \nabla u| \,dx\). The explicit form of the first variation of \(E\) with respect to \(L^2\) inner product is given by the degenerate parabolic equation \(u_t=\)div\(\, (\nabla u/| \nabla u| )\). More generaly, this paper deals with general flow systems for which the energy density is not \(C^1\), so that the diffusivity in the equation is very strong and its effect is even nonlocal. A first attempt in this paper is to formulate a constrained gradient system when the energy \(\phi\) is convex but having singularities. In this case the equation is formally written as \(u_t\in -P_u(\partial\phi (u))\) on some fixed Riemannian manifold \(M\), where \(P_u\) is the orthogonal projection to the tangent space of \(M\) at the value of \(u\). Next, it is studied the behaviour of the solution in the particular case \(M=S^1\). Several numerical simulations are presented in the last part of the present paper. Although the theoretical approach is restricted to the piecewise constant solutions at this point, this method also applies to calculating the evolution of solutions in more general cases.
scientific article; zbMATH DE number 2110032