A time-dependent obstacle problem in linearised elasticity
type: Item
instance of: scholarly article
title: A time-dependent obstacle problem in linearised elasticity
zbMATH Open document ID: 1445.35255
author: Paolo Piersanti
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2020-04-07T00:00:00Z
review text: The author deals with a unilateral hyperbolic problem for an elastic body moving against a plane obstacle. It has the following variational formulation: \begin{align*} &\text{Find}\ \boldsymbol{u}\equiv (u_i):(0,T)\to\boldsymbol{V}(\varOmega) \ \text{s.t.}\\& \boldsymbol{u}\in L^\infty(0,T;\boldsymbol{U}(\varOmega)),\ \dot{\boldsymbol{u}}\in L^\infty(0,T;\boldsymbol{L}^2(\varOmega)),\ \ddot{\boldsymbol{u}}\in \mathcal{M}([0,T];\boldsymbol{L}^2(\varOmega)), \\ & 2\rho\langle\langle\text{d}\ddot{\boldsymbol{u}}(t),\boldsymbol{v}(t)-\boldsymbol{u}(t)\rangle\rangle_{\boldsymbol{L}^2(\varOmega))}+\int_0^T \int_\varOmega A^{ijk\ell}e_{k||\ell}(\boldsymbol{u}(t))e_{i||j}(\boldsymbol{v}(t)-\boldsymbol{u}(t))\,\text{d}x\text{d}t\ge\\ & \int_0^T \int_\varOmega f^i(t)(v_i(t)-u_i(t))\,\text{d}x\text{d}t\ \forall \boldsymbol{v}\in \mathcal{D}(0,T;\boldsymbol{V}(\varOmega))\ \text{s.t.} \ \boldsymbol{v}(t)\in \boldsymbol{U}(\varOmega)\ \text{for a.e.}\ t\in (0,T)\\ &\text{and}\quad \boldsymbol{u}(0)=\boldsymbol{u}_0,\ \dot{\boldsymbol{u}}(0)=\boldsymbol{u}_1. \end{align*} The existence theorem is verified using the penalization method and rather strong assumption on the uniform estimate of penalized solutions.
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35L87
Mathematics Subject Classification ID: 35L86
Mathematics Subject Classification ID: 35Q74
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 7187775
zbMATH Keywords: penalty method
zbMATH Keywords: variational inequalities
zbMATH Keywords: plane obstacle
MaRDI profile type: Publication
full work available at URL: j.na.2019.111660
OpenAlex ID: W2951258533
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DOI: 10.1016/J.NA.2019.111660
description: scientific article; zbMATH DE number 7187775