Given an open, bounded and connected set Omega subset of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}&lt;^&gt;{3}$$\end{document} and its rescaling Omega epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\varepsilon }$$\end{document} of size epsilon &lt;&lt; 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll 1$$\end{document}, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation (epsilon-2 chi Omega epsilon+chi R3\Omega epsilon)partial derivative ttu=Delta u+f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} (\varepsilon &lt;^&gt;{-2}\chi _{\Omega _{\varepsilon }}+\chi _{\mathbb {R}&lt;^&gt;{3}\backslash \Omega _{\varepsilon }})\partial _{tt}u=\Delta u+f \end{aligned}\end{document}with initial data and source supported outside Omega epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\varepsilon }$$\end{document}; here, chi S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{S}$$\end{document} denotes the characteristic function of a set S. We provide the first-order epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the L infinity((0,1/epsilon tau),L2(R3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L&lt;^&gt;{\infty }((0,1/\varepsilon &lt;^&gt;{\tau }),L&lt;^&gt;{2}(\mathbb {R}&lt;^&gt;{3}))$$\end{document}-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in L2(Omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L&lt;^&gt;{2}(\Omega )$$\end{document} and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.

### The point scatterer approximation for wave dynamics

#### Abstract

Given an open, bounded and connected set Omega subset of R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}<^>{3}$$\end{document} and its rescaling Omega epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\varepsilon }$$\end{document} of size epsilon << 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll 1$$\end{document}, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation (epsilon-2 chi Omega epsilon+chi R3\Omega epsilon)partial derivative ttu=Delta u+f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} (\varepsilon <^>{-2}\chi _{\Omega _{\varepsilon }}+\chi _{\mathbb {R}<^>{3}\backslash \Omega _{\varepsilon }})\partial _{tt}u=\Delta u+f \end{aligned}\end{document}with initial data and source supported outside Omega epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\varepsilon }$$\end{document}; here, chi S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{S}$$\end{document} denotes the characteristic function of a set S. We provide the first-order epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the L infinity((0,1/epsilon tau),L2(R3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{\infty }((0,1/\varepsilon <^>{\tau }),L<^>{2}(\mathbb {R}<^>{3}))$$\end{document}-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in L2(Omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{2}(\Omega )$$\end{document} and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.
##### Scheda breve Scheda completa Scheda completa (DC)
2024
Wave equation; Point scatterer; Effective dynamics
Mantile, A.; Posilicano, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2176371
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