Low energy resolvent bounds for elliptic operators: An application to the study of waves in stratified media and fiber optics
- East Carolina Univ., Greenville, NC (United States)
For the positive self-adjoint operators H = -{partial_derivative}{sub igij}(x){partial_derivative} + q(x) on H {triple_bond} L{sup 2}(R{sup n}) with n {ge} 3, it is shown that {parallel}({vert_bar}x{vert_bar}){sup {minus}1}({radical}H - z){sup {minus}1}({vert_bar}x{vert_bar} + 1){sup {minus}1}{parallel} {le} C/Re(z){sup 2} as Re(z){yields}0{sup +}, by imposing several conditions on g{sub ij} and q. In the special case g{sub ij} = c{sup 2}{delta}{sub ij} these conditions reduce to {vert_bar}x{vert_bar}{center_dot}{del}c, (x{sup 2} + 1){sup 1+{var_epsilon}}({vert_bar}q(x){vert_bar} + {vert_bar}x{center_dot}{del}q{vert_bar}) {epsilon} L{sup {infinity}} with the nontrapping condition (c - x{center_dot} {Delta}c) {ge} kc, and a positivity condition C(x{sup 2} + 1){sup {minus}1} {le} 4p{vert_bar}x{vert_bar}{sup {minus}2} -(N - 2){sup 2}(-q){sub +}, for some k, C, p > 0. Results are applied to the stratified wave equation ({partial_derivative}{sub t}{sup 2} - c{sup 2}(y){Delta}{sub z}){psi} = 0, where z = x{circle_plus} y {epsilon} R{sup k} {circle_plus} R{sup m} with n = k + m, and {vert_bar}y{vert_bar}(y{center_dot}{del}{sub y}c){epsilon} L{sup {infinity}}(R{sup m}). In all cases the condition (c-y{center_dot}{del}{sub y}c) {le} kc leads to a local-energy decay estimate for {psi}(z,1). 11 refs.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 482472
- Journal Information:
- Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 11-12 Vol. 20; ISSN CPDIDZ; ISSN 0360-5302
- Country of Publication:
- United States
- Language:
- English
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