On local non-zero constraints in PDE with analytic coefficients
We consider the Helmholtz equation with real analytic coefficients on a bounded domain $\Omega\subset\mathbb{R}^{d}$. We take $d+1$ prescribed boundary conditions $f^{i}$ and frequencies $\omega$ in a fixed interval $[a,b]$. We consider a constraint on the solutions $u_{\omega}^{i}$ of the form $\ze...
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| Format: | Journal article |
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2015
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| _version_ | 1797053213306781696 |
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| author | Alberti, G Capdeboscq, Y |
| author_facet | Alberti, G Capdeboscq, Y |
| author_sort | Alberti, G |
| collection | OXFORD |
| description | We consider the Helmholtz equation with real analytic coefficients on a bounded domain $\Omega\subset\mathbb{R}^{d}$. We take $d+1$ prescribed boundary conditions $f^{i}$ and frequencies $\omega$ in a fixed interval $[a,b]$. We consider a constraint on the solutions $u_{\omega}^{i}$ of the form $\zeta(u_{\omega}^{1},\ldots,u_{\omega}^{d+1},\nabla u_{\omega}^{1},\ldots,\nabla u_{\omega}^{d+1})\neq0$, where $\zeta$ is analytic, which is satisfied in $\Omega$ when $\omega=0$. We show that for any $\Omega^{\prime}\Subset\Omega$ and almost any $d+1$ frequencies $\omega_{k}$ in $[a,b]$, there exist $d+1$ subdomains $\Omega_{k}$ such that $\Omega^{\prime}\subset\cup_{k}\Omega_{k}$ and $\zeta(u_{\omega_{k}}^{1},\ldots,u_{\omega_{k}}^{d+1},\nabla u_{\omega_{k}}^{1},\ldots,\nabla u_{\omega_{k}}^{d+1})\neq0$ in $\Omega_{k}$. This question comes from hybrid imaging inverse problems. The method used is not specific to the Helmholtz model and can be applied to other frequency dependent problems. |
| first_indexed | 2024-03-06T18:40:48Z |
| format | Journal article |
| id | oxford-uuid:0ccbe46d-e17e-4855-b860-945400113f52 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:40:48Z |
| publishDate | 2015 |
| record_format | dspace |
| spelling | oxford-uuid:0ccbe46d-e17e-4855-b860-945400113f522022-03-26T09:36:55ZOn local non-zero constraints in PDE with analytic coefficientsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:0ccbe46d-e17e-4855-b860-945400113f52Symplectic Elements at Oxford2015Alberti, GCapdeboscq, YWe consider the Helmholtz equation with real analytic coefficients on a bounded domain $\Omega\subset\mathbb{R}^{d}$. We take $d+1$ prescribed boundary conditions $f^{i}$ and frequencies $\omega$ in a fixed interval $[a,b]$. We consider a constraint on the solutions $u_{\omega}^{i}$ of the form $\zeta(u_{\omega}^{1},\ldots,u_{\omega}^{d+1},\nabla u_{\omega}^{1},\ldots,\nabla u_{\omega}^{d+1})\neq0$, where $\zeta$ is analytic, which is satisfied in $\Omega$ when $\omega=0$. We show that for any $\Omega^{\prime}\Subset\Omega$ and almost any $d+1$ frequencies $\omega_{k}$ in $[a,b]$, there exist $d+1$ subdomains $\Omega_{k}$ such that $\Omega^{\prime}\subset\cup_{k}\Omega_{k}$ and $\zeta(u_{\omega_{k}}^{1},\ldots,u_{\omega_{k}}^{d+1},\nabla u_{\omega_{k}}^{1},\ldots,\nabla u_{\omega_{k}}^{d+1})\neq0$ in $\Omega_{k}$. This question comes from hybrid imaging inverse problems. The method used is not specific to the Helmholtz model and can be applied to other frequency dependent problems. |
| spellingShingle | Alberti, G Capdeboscq, Y On local non-zero constraints in PDE with analytic coefficients |
| title | On local non-zero constraints in PDE with analytic coefficients |
| title_full | On local non-zero constraints in PDE with analytic coefficients |
| title_fullStr | On local non-zero constraints in PDE with analytic coefficients |
| title_full_unstemmed | On local non-zero constraints in PDE with analytic coefficients |
| title_short | On local non-zero constraints in PDE with analytic coefficients |
| title_sort | on local non zero constraints in pde with analytic coefficients |
| work_keys_str_mv | AT albertig onlocalnonzeroconstraintsinpdewithanalyticcoefficients AT capdeboscqy onlocalnonzeroconstraintsinpdewithanalyticcoefficients |