# A Bernstein property of solutions to a class of prescribed by McCoy J. A.

By McCoy J. A.

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If two positive quantities f and g satisﬁes A−1 f /g A with some constant A 1, then we say that f and g are comparable and write f ≈ g. We consider a pair (V, K) of a bounded open set V ⊂ Rn and a compact set K ⊂ Rn such that K ⊂ V , K ∩ D = ∅, and K ∩ ∂D = ∅. 1. 1) there exists a constant A0 depending only on D, V and K with the following property: If u and v are positive superharmonic functions on D such that (i) u and v are bounded, positive, and harmonic in V ∩ D, (ii) u and v vanish on V ∩ ∂D except for a polar set, then u(x)/u(y) v(x)/v(y) A0 for x, y ∈ K ∩ D.

1, no. : Some Applications of Functinal Analysis in Mathematical Physics. Third edition. Appendix. Am. Math. , Providence, RI (1991) 24. : On a theorem of functional analysis (Russian, French). Dokl. Akad. Nauk SSSR 20, no. 1, 5-9 (1938) 25. : On the Cauchy problem for quasilinear hyperbolic equations (Russian, French). Dokl. Akad. Nauk SSSR 20, no. 2-3, 79-83 (1938) 26. : On a theorem of functional analysis (Russian). Mat. Sb. 4, no. : In: Eleven Papers in Analysis. Am. Math. Soc. Transl. (2) 34, 39-68 (1963) 27.

L. Sobolev became very attractive and were developed by many mathematicians in diﬀerent ways. The second period could be dated from 1941 to 1957. L. Sobolev, was to evacuate the Institute from Moscow to Kazan’ and to organize conditions for work. L. Sobolev was appointed as the deputy director of the Institute for Atomic Energy. The best Soviet mathematicians were gathered under the roof of this institute to work on the nuclear project. L. Sobolev worked there for more than 14 years, however not much is known about these years.