Geodesic
An energy inequality and the overdeterminacy property of a system of eigenfunctions
Izvestiya. Mathematics , Tome 2 (1968) no. 4, pp. 695-707.

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In this article we prove an energy inequality for the third boundary value problem (the parabolic case) and certain completeness and closure properties of the system of eigenfunctions. 
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     author = {V. I. Plotnikov},
     title = {An energy inequality and the overdeterminacy property of a system of eigenfunctions},
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V. I. Plotnikov. An energy inequality and the overdeterminacy property of a system of eigenfunctions. Izvestiya. Mathematics , Tome 2 (1968) no. 4, pp. 695-707. http://geodesic.mathdoc.fr/item/IM2_1968_2_4_a1/

[1] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[2] Plotnikov V. I., “Teoremy edinstvennosti, suschestvovaniya i apriornye svoistva obobschennykh reshenii”, Dokl. AN SSSR, 165:1 (1965), 33–35 | MR | Zbl

[3] Smirnov V. I., Kurs vysshei matematiki, tom V, Fizmatgiz, M., 1960

[4] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, LGU, L., 1950