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Integral estimates of the solutions to the helmholtz equation in unbounded domains

Authors:

A. V. Filinovskii

Affiliation:

(1) N. é. Bauman Moscow State Technical University, Moscow, USSR

Abstract:

The following boundary value problem is studied:

$Delta v + omega ^2 v = hleft( x right), x in Omega subset mathbb{R}^n , n geqslant 2, - infty< omega< + infty , left. v right|_Gamma = 0, Gamma = partial Omega ;$

here the surface Г satisfies the condition( $left( {v,nabla varphi (x)} right)left| {_Gamma } right. leqslant 0$ , where

$varphi (x) = sumlimits_{j = 1}^n {alpha _j } x_j^2 , 0< alpha _1 leqslant alpha _1 leqslant cdots leqslant alpha _n = 1,$

and ν is the outward (with respect to Ω) normal to Γ. Translated fromMatematischskie Zametki, Vol. 61, No. 5, pp. 759–768, May, 1997. Translated by N. K. Kulman

Keywords:

Helmholtz equation boundary value problem integral estimate unbounded domain

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