Well-posedness and stability of the Lagrange representation of the n-D wave equation via boundary triples
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912763130937344 |
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| author | Aigner, Bernhard Skrepek, Nathanael |
| author_facet | Aigner, Bernhard Skrepek, Nathanael |
| contents | We study the Lagrange representation of the wave equation with generalized Laplacian $\operatorname{div} T \nabla$. We allow the coefficients -- the Young modulus $T$ and the density $ρ$ -- to be $\mathrm{L}^{\infty}$ or even nonlocal operators. Moreover, the Lipschitz boundary of the domain $Ω$ can be split into several parts admitting Dirichlet, Neumann and/or Robin-boundary conditions of displacement, velocity and stress. We show well-posedness of this classical model of the wave equation utilizing boundary triple theory for skew-adjoint operators. In addition we show semi-uniform stability of solutions under slightly stronger assumptions by means of a spectral result. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and stability of the Lagrange representation of the n-D wave equation via boundary triples Aigner, Bernhard Skrepek, Nathanael Analysis of PDEs Functional Analysis 35L05, 35B35, 46N20, 47B02, 47B44 We study the Lagrange representation of the wave equation with generalized Laplacian $\operatorname{div} T \nabla$. We allow the coefficients -- the Young modulus $T$ and the density $ρ$ -- to be $\mathrm{L}^{\infty}$ or even nonlocal operators. Moreover, the Lipschitz boundary of the domain $Ω$ can be split into several parts admitting Dirichlet, Neumann and/or Robin-boundary conditions of displacement, velocity and stress. We show well-posedness of this classical model of the wave equation utilizing boundary triple theory for skew-adjoint operators. In addition we show semi-uniform stability of solutions under slightly stronger assumptions by means of a spectral result. |
| title | Well-posedness and stability of the Lagrange representation of the n-D wave equation via boundary triples |
| topic | Analysis of PDEs Functional Analysis 35L05, 35B35, 46N20, 47B02, 47B44 |
| url | https://arxiv.org/abs/2505.08602 |