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Disoscillability of solutions to partial differential equations on manifolds of constant curvature and mean value theorems. (English. Russian original) Zbl 0880.35010
Sib. Math. J. 38, No. 5, 871-880 (1997); translation from Sib. Mat. Zh. 38, No. 5, 1008-1017 (1997).
The mean value theorem plays an important role in the theory of harmonic functions. In the article under review, some analogs of this theorem in the spaces of constant curvature are derived for the polyharmonic equation \[ L^{n}u + c_1 L^{n-1}u + \ldots + c_n u = 0.\tag{1} \] Here \(L\) is the Laplace-Beltrami operator. The mean value \[ M_r \left[ u(x), P_0 \right] = \frac{1}{A(r)} \int \ldots \int\limits_{S_r} u(x) ds \] is introduced, where \(A(r)\) is the norming factor of the space. The author calls a nontrivial solution to (1) oscillating in a domain \(D\) if its mean value \(M(r)\) is an oscillating function at least for one point \(P_0 \in D\), i.e., it has at least \(2n\) zeroes. Some theorems are proven stating necessary and sufficient conditions for disoscillability of the solution in terms of the function \(M(r)\).
MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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