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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2606.01383 |
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| _version_ | 1866914621630185472 |
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| author | Avdonin, Sergei Edward, Julian |
| author_facet | Avdonin, Sergei Edward, Julian |
| contents | Let $Δ$ be the Dirichlet Laplacian on a bounded domain $Ω\subset \mathbb{R}^{N}$, and let $(-Δ)^α$ be the associated spectral fractional Laplacian with $α\leq 1, \ ρ<2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $α<1/2$, extending a result previously proven on rectangles for $α<1$. As a consequence of this formula, well-posedness results are proven for the structurally damped plate equation $$u_{tt}+Δ^2u+(-Δ)^αu_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $α<1$, we prove boundary null-controllability results.
For $α<1/2, \ ρ\leq 2$, Dirichlet null controllability is proved for the unit disk in $\mathbb{R}^2$. This analysis then extended to the classical case, $α=1$, on rectangles, where higher regularity is required for Dirichlet control. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01383 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A symmetry formula for the spectral fractional Laplacian, and applications to boundary controllability for plate equation with structural damping Avdonin, Sergei Edward, Julian Analysis of PDEs Optimization and Control 93B05, 26A33, 74K10, 93C20 Let $Δ$ be the Dirichlet Laplacian on a bounded domain $Ω\subset \mathbb{R}^{N}$, and let $(-Δ)^α$ be the associated spectral fractional Laplacian with $α\leq 1, \ ρ<2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $α<1/2$, extending a result previously proven on rectangles for $α<1$. As a consequence of this formula, well-posedness results are proven for the structurally damped plate equation $$u_{tt}+Δ^2u+(-Δ)^αu_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $α<1$, we prove boundary null-controllability results. For $α<1/2, \ ρ\leq 2$, Dirichlet null controllability is proved for the unit disk in $\mathbb{R}^2$. This analysis then extended to the classical case, $α=1$, on rectangles, where higher regularity is required for Dirichlet control. |
| title | A symmetry formula for the spectral fractional Laplacian, and applications to boundary controllability for plate equation with structural damping |
| topic | Analysis of PDEs Optimization and Control 93B05, 26A33, 74K10, 93C20 |
| url | https://arxiv.org/abs/2606.01383 |