zbMATH — the first resource for mathematics

Investigation of the oscillatory properties of solutions to partial differential equations using sphere means. (English. Russian original) Zbl 0967.35036
Dokl. Math. 57, No. 3, 413-415 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 360, No. 4, 448-450 (1998).
From the introduction: In the investigation of the oscillation properties of solutions to elliptical partial differential equations with the Laplacian in the principal term, a sphere mean of the form \[ M_r[u(x), P_0]= {1\over \omega_n r^{n-1}} \int\underset{S_r}\cdots\int u(x) ds \] has been widely used since the middle 1970s. Here, we use this approach to investigate the oscillation properties of partial differential equations in spaces \(X\) of constant curvature of the form \[ L^nu+ p(x)u= 0, \] where \(L\) is the Laplace-Beltrami operator having different form for each particular space \(X\) of constant curvature.
35J15 Second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs