
This article is cited in 2 scientific papers (total in 2 papers)
Overdetermined Neumann boundary value problem in unbounded domains
V. V. Volchkov^{}, Vit. V. Volchkov^{} ^{} Donetsk National University,
Universitetskaya str. 24,
83001, Donetsk
Abstract:
The studying of overdetermined
boundary value problems for elliptic partial differential equations was initiated by J. Serrin in 1971. In his work, he established a property of radial symmetry for solutions of some overdetermined Poisson problem. Apart of a significant independent interest, the problems of such kind have important applications in the potential theory, integral geometry, hydrodynamics and capillarity theory. Usually, the resolving of these problems is based on Hopf lemma on an angular boundary point and the method of hyperplanes motion introduced by A.A. Alexandrov for studying some geometric problems related with characterizing the spheres. Among other more modern methods not involving the maximum principle for the considered problems we mention the duality method, the method of volume derivative as well as an integral method.
In the present paper we consider an overdetermined Neumann problem for the Laplace equation $\Delta f=0$ in planar unbounded domains. We show that under some conditions, see Theorem 1 in Section 1, such problem is solvable only for the exterior of a ball. A specific feature of Theorem 1 is that in this theorem, for the first time, we obtain an exact condition for the growth of $f$ at infinity. Moreover, as Theorem 2 in Section 2 shows, other conditions in Theorem 1 are also necessary. In contrast to the earlier works, the proof of Theorem 1 employs some boundary properties of conformal mappings, Smirnov theorem on functions in a class $H_p$ and FejerRiesz theorem on nonnegative trigonometrical polynomials.
Keywords:
overdetermined problems, Neumann problem, harmonic functions, boundary behavior.
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English version:
Ufa Mathematical Journal, 2020, 12:2, 10–20 (PDF, 364 kB); https://doi.org/10.13108/202012210
Bibliographic databases:
UDC:
517.5
MSC: 31A05 Received: 30.10.2019
Citation:
V. V. Volchkov, Vit. V. Volchkov, “Overdetermined Neumann boundary value problem in unbounded domains”, Ufimsk. Mat. Zh., 12:2 (2020), 10–20; Ufa Math. J., 12:2 (2020), 10–20
Citation in format AMSBIB
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\pages 1020
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\jour Ufa Math. J.
\yr 2020
\vol 12
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\pages 1020
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V. D. Gordevskii, O. V. Uvarov, I. Yu. Chudinovich, V. A. Shcherbina, “Bogolyubov's $R$operation for local interactions”, Theoret. and Math. Phys., 20:2 (1974), 729–737

I. Yu. Chudinovich, V. A. Shcherbina, “Renormalized feynman amplitudes for fields with fixed masses”, Theoret. and Math. Phys., 27:1 (1976), 307–315

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