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Morrey-Campanato estimates for Helmholtz equations. (English) Zbl 0932.35048

The uniform weighted \(L^{2}\) and Morrey-Campanato type estimates are obtained for the Helmholtz equation \(\Delta u+n(x)u=-f(x)\), \(x\in \mathbb{R}^{d}\), \(d\geq 2\), where \(n(x)>0\) is taken to be a variable ”index”. This Morrey-Campanato estimate generalizes some previous results [for instance S. Agmon and L. Hörmander, J. Anal. Math. 30, 1-38 (1976; Zbl 0335.35013)] from the case as \(n(x)\) is a perturbation of a constant to the case as \(n(x)\) is a variable ”index”. It is convenient that obtained uniform \(L^{2}\) estimate plays a fundamental role in solving Schrödinger evolution equation with nonlinear first order term [see C. Kenig, G. Ponce and L. Vega, Invent. Math. 134, 489-545 (1998; Zbl 0928.35158)]. The proof is based on a multiplier method which is an improvement of that used for the wave, Schrödinger or kinematic equation by C. S. Morawetz [Proc. R. Soc. Lond., Ser. A 306, 291-296 (1968; Zbl 0157.41502)], J. E. Lin and W. A. Strauss [J. Funct. Anal. 30, 245-263 (1978; Zbl 0395.35070)], and P. L. Lions and B. Perthame [C. R. Acad. Sci., Paris, Sér. I, 314, No. 11, 801-806 (1992; Zbl 0761.35085)].

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B45 A priori estimates in context of PDEs
35P25 Scattering theory for PDEs
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