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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
##### Keywords:
polyharmonic equation; Laplace-Beltrami operator
Full Text:
##### References:
 [1] R. Courant, Partial Differential Equations [Russian translation], Mir, Moscow (1964). · Zbl 0121.07801 [2] J. A. Wolf, Spaces of Constant Curvature [Russian translation], Nauka, Moscow (1982). [3] S. Helgason, The Radon Transform [Russian translation], Mir, Moscow (1983). · Zbl 0532.43008 [4] M. K. Bugir, Oscillation Properties of Solutions to Partial Differential Equations on Manifolds of Constant Curvature, [in Russian] [Preprint, No. 27], Akad. Nauk Ukrain. SSR, Inst. Mat. (Kiev), Kiev (1991). · Zbl 0765.35057 [5] T. Kusano and M. Naito, ”Oscillation criteria for a class of perturbed Schrödinger equations,” Canad. Math. Bull.,25, No. 1, 71–77 (1982). · Zbl 0488.35007 · doi:10.4153/CMB-1982-010-3 [6] Y. Kitamura and T. Kusano, ”Nonlinear oscillation of a fourth-order elliptic equation,” J. Differential Equations,30, No. 2, 280–286 (1978). · Zbl 0422.35004 · doi:10.1016/0022-0396(78)90017-7 [7] E. S. Noussair and C. A. Swanson, ”Oscillation theory for semilinear Schrödinger equations and inequalities,” Proc. Roy. Soc. Edinburgh Sect. A,75, No. 1, 67–81 (1975/76). · Zbl 0372.35004 [8] M. K. Bugir, ”Mean value theorems and disoscillability of solutions to partial differential equations in the spacesE n andP n,” Ukrain. Mat. Zh.,44, No. 11, 1477–1483 (1992). · Zbl 0799.35020 [9] H. Bateman and A. Erdélyi, Higher Transcendental Functions. Vol. 1: The Hypergeometric Function. Legendre Functions [Russian translation], Nauka, Moscow (1965). · Zbl 0146.09301 [10] V. G. Korenev, Introduction to the Theory of Bessel Functions [in Russian], Nauka, Moscow (1977). · Zbl 0357.65071 [11] M. K. Bugir, ”Disoscillability of solutions to systems of strongly elliptic differential equations,” Dopov. Akad. Nauk Ukrain. SSR Ser. A, No. 2, 101–105 (1975). · Zbl 0301.35008 [12] O. I. Bobik, P. I. Bondarchuk, B. I. Ptashnik, and V. Ya. Skorobogat’ko, Elements of the Qualitative Theory of Partial Differential Equations [in Ukrainian], Naukova Dumka, Kiev (1972). [13] M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskiî, Ill-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Moscow (1980). [14] V. A. Kondrat’ev, ”On oscillability of solutions to the equationsx (n)+p(t)x=0,” Trudy Moskov. Mat. Obshch.,10, 419–436 (1961). [15] M. K. Bugir, ”On disoscillability conditions for solutions to systems of differential equations in spaces of constant curvature,” in: Nonlinear Boundary Value Problems of Mathematical Physics and Their Applications [in Russian], Inst. Mat. (Kiev), Kiev, 1993, pp. 31–33. (Trudy Inst. Mat. Akad. Nauk Ukrain. SSR.)
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