Fixed Points of Multi-valued Maps and Static Coulomb Friction Problems |
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Authors: | Patrick J Rabier Ovidiu V Savin |
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Affiliation: | (1) Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, U.S.A.;(2) Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, U.S.A. |
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Abstract: | The existence of solutions to the Signorini problem with Coulomb friction is a long standing open question. We prove the existence
of generalized solutions that satisfy the pointwise Coulomb friction conditions on the entire interface and the normal nonpenetration
condition on the complement of a subset with arbitrarily small but possibly positive measure. Furthermore, the penetration
itself can also be made arbitrarily small. Although “measure zero” instead of “arbitrarily small measure” would be needed
to fully resolve the issue, these generalized solutions seem to be the closest answer available to date. Their existence is
proved by a suitable application of Ky Fan's fixed point theorem for multi-valued maps. The same method can be used with a
number of variants involving contact of two or more elastic bodies and possible debonding phenomena.
This revised version was published online in August 2006 with corrections to the Cover Date. |
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Keywords: | Coulomb friction multi-valued mapping Ky Fan theorem |
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