# American Institute of Mathematical Sciences

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A convex-concave elliptic problem with a parameter on the boundary condition
April  2012, 32(4): 1125-1167. doi: 10.3934/dcds.2012.32.1125

## Second order approximations of quasistatic evolution problems in finite dimension

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Received  November 2010 Revised  April 2011 Published  October 2011

In this paper, we study the limit, as $\epsilon$ goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^{\epsilon}(t)+\epsilon B\dot u^{\epsilon}(t)+\nabla_xf(t,u^{\epsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\epsilon}(t)$ 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.
Citation: Virginia Agostiniani. Second order approximations of quasistatic evolution problems in finite dimension. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125
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