Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908928149815296 |
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| author | Dias, João Paulo Neves, Wladimir Rodrigues, José Francisco |
| author_facet | Dias, João Paulo Neves, Wladimir Rodrigues, José Francisco |
| contents | We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two obstacles problem. In contrast with the linear setting, for both strictly quasilinear cases we obtain continuous solutions in a weak complementary sense, which moreover satisfy a weak entropy condition in the free region where the string is not in contact with the obstacles. We further show that, in both the one and two obstacle cases, these solutions are variational solutions in a hyperbolic sense without the viscosity term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00336 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation Dias, João Paulo Neves, Wladimir Rodrigues, José Francisco Analysis of PDEs 35L70, 35R35, 35L86 We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two obstacles problem. In contrast with the linear setting, for both strictly quasilinear cases we obtain continuous solutions in a weak complementary sense, which moreover satisfy a weak entropy condition in the free region where the string is not in contact with the obstacles. We further show that, in both the one and two obstacle cases, these solutions are variational solutions in a hyperbolic sense without the viscosity term. |
| title | Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation |
| topic | Analysis of PDEs 35L70, 35R35, 35L86 |
| url | https://arxiv.org/abs/2604.00336 |