Obstacle problem for von Kármán equations
type: Item
instance of: scholarly article
title: Obstacle problem for von Kármán equations
zbMATH Open document ID: 0766.35087
DOI: 10.1016/0196-8858(92)90005-H
author: Yang Gao
author: Shing-Tung Yau
published in: Advances in Applied Mathematics
publication date: 1993-01-16T00:00:00Z
review text: This work is devoted to the study of free boundary variational problems in the theory of thin elastic plates with obstacles. In the formulation of this free boundary value problem a plate is considered in possible contact with a rigid obstacle such that the plate is subjected to a distributed load and the shape of the obstacle is given by a prescribed strictly concave function that belongs to some Sobolev space. The boundary of the set of points for which the vertical displacements coincide with the shape of the obstacle (the free boundary) is a priori unknown. A special nonlinear relation is also considered between the strain tensor and deformations, and usual symmetry and ellipticity properties for the elasticity tensor are assumed. The authors prove that any solution of the former problem fulfills some variational inequality, but in the geometric nonlinear case they prove that the variational inequality is not equivalent to the corresponding variational extreme problem. A complementary method is developed here in order to show that the primal nonlinear variational problem is equivalent to a semiquadratic dual optimization problem defined on the statically admissible space. Here the Rockafellar's duality theory is generalized. The authors prove that the uniqueness and existence of the solution depend directly on the gap energy function.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 74A55
Mathematics Subject Classification ID: 74M15
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 74P10
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74K20
zbMATH DE Number: 91493
zbMATH Keywords: free boundary variational problems
zbMATH Keywords: thin elastic plates with obstacles
zbMATH Keywords: rigid obstacle
zbMATH Keywords: primal nonlinear variational problem
zbMATH Keywords: semiquadratic dual optimization problem
zbMATH Keywords: Rockafellar's duality theory
zbMATH Keywords: uniqueness
zbMATH Keywords: existence
reviewed by: Mariano Rodríguez Ricard
MaRDI profile type: Publication
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full work available at URL: 0196-8858(92)90005-h
OpenAlex ID: W2055249833
description: scientific article; zbMATH DE number 91493