Calculus of Variations and Geometric Measure Theory
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L. Nardini

A note on the convergence of singularly perturbed second order potential-type equations

created by nardini on 11 Mar 2015
modified on 22 May 2017

[BibTeX]

Published Paper

Inserted: 11 mar 2015
Last Updated: 22 may 2017

Journal: Journal of Dynamics and Differential Equations
Volume: 29
Number: 2
Pages: 783-797
Year: 2017
Doi: 10.1007/s10884-015-9461-y

Abstract:

In this paper we study the limit as $\varepsilon \to 0$ of the second order equation $\varepsilon^2 \ddot{u}_\varepsilon + \nabla_{\!x} V(t,u_\varepsilon(t))=0$, where $V(t,x)$ is a potential. We assume that $u_0(t)$ is one of its equilibrium points such that $\nabla_{\!x}V(t,u_0(t))=0$ and $\nabla_{\!x}^2V(t,u_0(t))>0$. We find that, under suitable initial data, the solutions $u_\varepsilon$ converge uniformly to $u_0$, by imposing mild hypotheses on $V$. A counterexample shows that they can not be weakened

Keywords: singular perturbations, Vanishing inertia, Dynamic solutions


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