A variational inequality for total variation functional with constraint (Q5947222)

The authors study a nonlinear inclusion of the form NEWLINE\[NEWLINE\kappa \partial V(w)\ni w + \theta_0\quad \text{in}\quad L^2(0,L),\tag{1}NEWLINE\]NEWLINE where \(L\) is a positive and finite number, \(\theta_0\) is a given constant, \(\kappa\) is a (small) positive constant and \(V(w)\) is the total variation of any function \(w\) with \(|w|\leq 1\) on \([0,L]\). As easily seen, (1) is the Euler-Lagrange inclusion of the functional NEWLINE\[NEWLINEF_{\theta_0}(z):=\kappa V(z)-\frac{1}{2}\int_0^Lz^2 dx-\int_0^L\theta_0 z dx\quad\text{on}\quad L^2(0,L).NEWLINE\]NEWLINE The structure of the solution set of (1) for each prescribed constant \(\theta_0\) is studied and the expression for each solution \(w\) is given. It is shown that any solution \(w\) of (1) is piecewise constant and has a finite number of discontinuities in \([0,L]\). As an application, a more detailed characterization of the local minimizers of \(F_{\theta_0}\) in the one-dimensional case is obtained.