arXiv subjects
Publications and source records attributed to Augusto Visintin.
A notion of evolutionary $Γ$-convergence of weak type is introduced for sequences of operators acting on time-dependent functions. This extends the classical definition of $Γ$-convergence of functionals due to De Giorgi. The $Γ$-compactness of equi-coercive and equi-bounded sequences of operators is proved. Applications include the structural compactness and stability of quasilinear flows for pseudo-monotone operators.
Fitzpatrick's variational representation of maximal monotone operators is here extended to a class of pseudo-monotone operators in Banach spaces. On this basis, the initial-value problem associated with the first-order flow of such an operator is here reformulated as a minimization principle, extending a method that was pioneered by Brezis, Ekeland and Nayroles for gradient flows. This formulation is used to prove that the problem is stable w.r.t.\ arbitrary perturbations not only of data but also of operators. This is achieved by using the notion of evolutionary $Γ$-convergence w.r.t.\ a nonlinear topology of weak type. These results are applied to the Cauchy problem for quasilinear parabolic PDEs. This provides the structural compactness and stability of the model of several physical phenomena: nonlinear diffusion, incompressible viscous flow, phase transitions, and so on.
The initial-value problem associated with multi-valued operators in Banach spaces is here reformulated as a minimization principle, extending results of Brezis-Ekeland, Nayroles and Fitzpatrick. At the focus there is the stability of these problems w.r.t.\ perturbations not only of data but also of operators; this is achieved via De Giorgi's $Γ$-convergence w.r.t.\ a nonlinear topology of weak type. A notion of evolutionary $Γ$-convergence of weak type is also introduced. These results are applied to quasilinear PDEs, including doubly-nonlinear flows.
This work deals with the homogenization of an initial- and boundary-value problem for the doubly-nonlinear system $$ D_t w -\nabla\cdot \vec z = \nabla\cdot \vec h(x,t,x/\varepsilon), \qquad w\in α(u,x/\varepsilon), \qquad \vec z\in \vecγ(\nabla u,x/\varepsilon). $$ Here $\varepsilon$ is a positive parameter, and the prescribed mappings $α$ and $\vecγ$ are maximal monotone with respect to the first variable and periodic with respect to the second one. The two inclusions are here formulated as null-minimization principles, via the theory of Fitzpatrick [MR 1009594]. As $\varepsilon\to 0$, a two-scale formulation is derived via Nguetseng's notion of two-scale convergence, and a (single-scale) homogenized problem is then retrieved.