Sensitivity analysis of a Signorini-type history-dependent variational inequality
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912317214556160 |
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| author | Betz, Livia Matei, Andaluzia Sofonea, Mircea |
| author_facet | Betz, Livia Matei, Andaluzia Sofonea, Mircea |
| contents | We consider a history-dependent variational inequality (P) which models the frictionless contact between a viscoelastic body and a rigid obstacle covered by a layer of soft material. The inequality is expressed in terms of the displacement field, is governed by the data f (related to the applied body forces and surface tractions) and, under appropriate assumptions, it has a unique solution, denoted by u. Our aim in this paper is to perform a sensitivity analysis of the inequality (P), including the study of the regularity of the solution operator $f \mapsto u$. To this end, we start by proving the equivalence of (P) with a fixed point problem, denoted by (Q). We then consider an associated optimal control problem, for which we present an existence result. Then, we prove the directional differentiability of the solution operator and show that the directional derivative is characterized by a history-dependent variational inequality with time-dependent constraints. Finally, we prove two well-posedness results in the study of problems (P) and (Q), respectively, and compare the two well-posedness concepts employed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sensitivity analysis of a Signorini-type history-dependent variational inequality Betz, Livia Matei, Andaluzia Sofonea, Mircea Analysis of PDEs 49J40, 47J20, 74M15, 47G10, 49J20, 49K50 We consider a history-dependent variational inequality (P) which models the frictionless contact between a viscoelastic body and a rigid obstacle covered by a layer of soft material. The inequality is expressed in terms of the displacement field, is governed by the data f (related to the applied body forces and surface tractions) and, under appropriate assumptions, it has a unique solution, denoted by u. Our aim in this paper is to perform a sensitivity analysis of the inequality (P), including the study of the regularity of the solution operator $f \mapsto u$. To this end, we start by proving the equivalence of (P) with a fixed point problem, denoted by (Q). We then consider an associated optimal control problem, for which we present an existence result. Then, we prove the directional differentiability of the solution operator and show that the directional derivative is characterized by a history-dependent variational inequality with time-dependent constraints. Finally, we prove two well-posedness results in the study of problems (P) and (Q), respectively, and compare the two well-posedness concepts employed. |
| title | Sensitivity analysis of a Signorini-type history-dependent variational inequality |
| topic | Analysis of PDEs 49J40, 47J20, 74M15, 47G10, 49J20, 49K50 |
| url | https://arxiv.org/abs/2504.06402 |