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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 and Continuous Dynamical Systems, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125
##### References:
 [1] F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071. doi: 10.1142/S0218202508002942. [2] G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5. [3] G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, Calc. Var. Partial Differential Equations, 40 (2011), 125-181. [4] G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution, SISSA preprint 46/2010/M. [5] G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case, Netw. Heterog. Media, 5 (2010), 97-132. [6] M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167. [7] J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. [8] J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. [9] M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. [10] D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Phys. D, 239 (2010), 1470-1484. doi: 10.1016/j.physd.2009.02.008. [11] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585. [12] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 2011. doi: 10.1051/cocv/2010054. [13] F. Solombrino, Quasistatic evolution for plasticity with softening: the spatially homogeneous case, Discrete Contin. Dyn. Syst., 27 (2010), 1189-1217. doi: 10.3934/dcds.2010.27.1189. [14] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), 2 (2009), 1-35. [15] C. Zanini, Singular perturbation of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.

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##### References:
 [1] F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path, Math. Models Methods Appl. Sci., 18 (2008), 1027-1071. doi: 10.1142/S0218202508002942. [2] G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5. [3] G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, Calc. Var. Partial Differential Equations, 40 (2011), 125-181. [4] G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution, SISSA preprint 46/2010/M. [5] G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case, Netw. Heterog. Media, 5 (2010), 97-132. [6] M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167. [7] J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. [8] J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. [9] M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. [10] D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Phys. D, 239 (2010), 1470-1484. doi: 10.1016/j.physd.2009.02.008. [11] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585. [12] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 2011. doi: 10.1051/cocv/2010054. [13] F. Solombrino, Quasistatic evolution for plasticity with softening: the spatially homogeneous case, Discrete Contin. Dyn. Syst., 27 (2010), 1189-1217. doi: 10.3934/dcds.2010.27.1189. [14] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9), 2 (2009), 1-35. [15] C. Zanini, Singular perturbation of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.

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