Calculus of Variations and Geometric Measure Theory

V. Agostiniani

Second order approximations of quasistatic evolution problems in finite dimension

created by virginia on 15 Dec 2010
modified on 15 Apr 2011


Accepted Paper

Inserted: 15 dec 2010
Last Updated: 15 apr 2011

Journal: Discrete Contin. Dyn. Syst.
Year: 2010


In this paper, we study the limit, as $\varepsilon$ goes to zero, of a particular solution of the equation ${\varepsilon}^2A{\ddot u}^{\varepsilon}(t)+\varepsilon B{\dot u}^{\varepsilon}(t)+\nabla_xf(t,u^{\varepsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\varepsilon}(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.

Keywords: singular perturbations, perturbation methods, discrete approximations, vanishing viscosity, saddle-node bifurcation, heteroclinic solutions