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Register a new userOptimal constants in nontrapping resolvent estimates and applications in numerical analysis
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Added by
Euan Spence
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $|chi R(k) chi|_{L^2to L^2}leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/pi$ times the length of the longest generalized bicharacteristic of $|xi|_g^2-1$ remaining in the support of $chi.$ We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation.
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We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form $|psi(| abla|) exp(itphi(| abla|)f |_{L^2(w)} leq C|f|_{L^2}$, where the weight $w$ is radial and depends only on the spatial variable; such a smoothing estimate is of course equivalent to the $L^2$-boundedness of a certain oscillatory integral operator $S$ depending on $(w,psi,phi)$. Furthermore, when $w$ is homogeneous, and for certain $(psi,phi)$, we provide an explicit spectral decomposition of $S^*S$ and consequently recover an explicit formula for the optimal constant $C$ and a characterisation of extremisers. In certain well-studied cases when $w$ is inhomogeneous, we obtain new expressions for the optimal constant.
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