Strong asymptotic convergence of evolution equations governed by maximal monotone operators

We consider the Tikhonov-like dynamics $-\dot u(t)\in A(u(t))+\varepsilon(t)u(t)$ where $A$ is a maximal monotone operator and the parameter function $\eps(t)$ tends to 0 for $t\to\infty$ with $\int_0^\infty\eps(t)dt=\infty$. When $A$ is the subdifferential of a closed proper convex function $f$, we establish strong convergence of $u(t)$ towards the least-norm minimizer of $f$. In the general case we prove strong convergence towards the least-norm point in $A^{-1}(0)$ provided that the function $\eps(t)$ has bounded variation, and provide a counterexample when this property fails.


Journal of Differential Equations, Vol 245 (2008), 3753-3763.