Morrey-Campanato estimates for Helmholtz equations
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 \textit{S. Agmon} and \textit{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 \textit{C. Kenig, G. Ponce} and \textit{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 \textit{C. S. Morawetz} [Proc. R. Soc. Lond., Ser. A 306, 291-296 (1968; Zbl 0157.41502)], \textit{J. E. Lin} and \textit{W. A. Strauss} [J. Funct. Anal. 30, 245-263 (1978; Zbl 0395.35070)], and \textit{P. L. Lions} and \textit{B. Perthame} [C. R. Acad. Sci., Paris, Sér. I, 314, No. 11, 801-806 (1992; Zbl 0761.35085)].
scientific article; zbMATH DE number 1309117