Local solvability of a constrained gradient system of total variation
The authors study the initial value problem for the total variation flow of maps from a domain \(\Omega\subset\mathbb{R}^n\) into a closed submanifold \(M\subset\mathbb{R}^N\). Formally, this flow equation can be written as \(u_t= \pi_k(\text{div}(\nabla u/|\nabla u|))\), where \(\pi_y\) denotes the orthogonal projection from \(\mathbb{R}^N\) onto the tangent space of \(M\) at \(y\in M\). The precise meaning of this equation is given in terms of the subdifferential \(\partial\Phi^T\) of the proper, convex energy functional \(\Phi^T(u)= \int^T_0(\int|\nabla u(x)|\,dx)\,dt\) on the space \(L^2(0,T; L^2(\Omega, \mathbb{R}^N))\) and reads \(u_t\in-P_u(\partial\Phi^T(u))\) where \(P_u(f)(x, t))= \pi_{u(x,t)}(f(x,t))\). The authors develop their theory in the case of periodic boundary conditions (i.e. \(\Omega\) is replaced by the torus \(\mathbb{T}^n\)) as well as in the case of a Dirichlet boundary condition, but the main result, a local existence theorem, refers only to the case of periodic boundary conditions. This solution is obtained as a limit of solutions of the \(p\)-harmonic map flow equation where \(p= 1+ 1/m\) and \(m\) tends to infinity. The existence of the latter solutions for smooth initial data of small \(p\)-energy (depending also on the Lipschitz constant of the initial values) was established by \textit{A. Fardoun} and \textit{R. Regbaoui} [Calc. Var. Partial Differ. Equ. 16, No. 1, 1--16 (2003; Zbl 1034.58013)]. Using this result, the authors are able to prove the existence of a solution \(u\in C([0, T], L^2(\mathbb{T}^n, M))\), solving the total variation flow equation in the \(L^2(0,T; L^2(\mathbb{T}^n, \mathbb{R}^N))\)-sense provided that (essentially) the \(1+{1\over m_0}\)-energy of the \(C^{2+\alpha}\)-smooth initial data is sufficiently small for some sufficiently large \(m_0\in\mathbb{N}\), the time interval \([0, T]\) also depending on the Lipschitz constant of the initial values.
scientific article; zbMATH DE number 2163681