Morrey–Campanato Estimates for Helmholtz Equations

Abstract We derive uniform weighted L 2 and Morrey–Campanato type estimates for Helmholtz equations in a medium with a variable index which is not necessarily constant at infinity. Our technique is based on a multiplier method with appropriate weights which generalize those of Morawetz for the wave equation. We also extend our method to the wave equation.

[1]  P. Markowich,et al.  Homogenization limits and Wigner transforms , 1997 .

[2]  Jean-David Benamou,et al.  High frequency limit of the Helmholtz equations. , 2002 .

[3]  C. Bardos,et al.  Scattering frequencies and Gervey 3 singularities , 1987 .

[4]  Lemmes de moments, de moyenne et de dispersion , 1992 .

[5]  Barry Simon,et al.  Analysis of Operators , 1978 .

[6]  H. Brezis,et al.  Quantization effects for −Δu = u(1 − |u|2) in ℝ2 , 1994 .

[7]  Luis Vega,et al.  Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations , 1998 .

[8]  Shmuel Agmon,et al.  Asymptotic properties of solutions of differential equations with simple characteristics , 1976 .

[9]  T. Colin.  Smoothing effects for dispersive equations via a generalized Wigner transform , 1994 .

[10]  L. Vega,et al.  Weighted Estimates for the Helmholtz Equation and Some Applications , 1997 .

[11]  Cathleen S. Morawetz,et al.  Time decay for the nonlinear Klein-Gordon equation , 1968, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[12]  Luis Vega,et al.  Small solutions to nonlinear Schrödinger equations , 1993 .

[13]  Ingenuin Gasser,et al.  Dispersion and Moment Lemmas Revisited , 1999 .

[14]  T. Paul,et al.  Sur les mesures de Wigner , 1993 .

[15]  B. Zhang,et al.  Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  W. Strauss,et al.  Decay and scattering of solutions of a nonlinear Schrödinger equation , 1978 .