Geodesic
Variational Problem with Degeneracy at the Boundary of the Interval
Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 464-471.

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The weighted $L_p$-norms of derivatives are estimated in terms of the weighted $L_p$-norm of the highest derivative and the traces of the function and its derivatives at the given points of closure of the bounded interval; weights are powers of the distance to the nearest endpoint of the interval. For functions with zero traces, sharper estimates are established. For the integral quadratic functional with degenerate coefficients, we prove the existence and uniqueness of the solution to the problem of minimization of a functional on a function class with zero traces.
Keywords: variational problem, weighted $L_p$-norm, minimization of a functional, function class with zero traces, Hardy's inequality, Wronskian. 
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V. V. Shan'kov. Variational Problem with Degeneracy at the Boundary of the Interval. Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 464-471. http://geodesic.mathdoc.fr/item/MZM_2007_81_3_a13/

[1] S. M. Nikolskii, P. I. Lizorkin, “O nekotorom neravenstve dlya funktsii iz vesovykh klassov i kraevykh zadachakh s silnym vyrozhdeniem na granitse”, Dokl. AN SSSR, 159:3 (1964), 512–515 | MR | Zbl

[2] S. M. Nikolskii, Priblizheniya funktsii mnogikh deistvitelnykh peremennykh i teoremy vlozheniya, 2-e izd., Nauka, M., 1977 | MR | Zbl

[3] S. M. Nikolskii, “Variatsionnaya problema dlya uravneniya ellipticheskogo tipa s vyrozhdeniem na granitse”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. VII, Tr. MIAN, 150, 1979, 212–238 | MR | Zbl

[4] P. I. Lizorkin, S. M. Nikolskii, “Koertsitivnye svoistva ellipticheskogo uravneniya s vyrozhdeniem. Variatsionnyi metod”, Teoriya chisel, matematicheskii analiz i ikh prilozheniya, Tr. MIAH, 157, l981, 90–118 | MR | Zbl

[5] L. D. Kudryavtsev, “O suschestvovanii i edinstvennosti reshenii variatsionnykh zadach”, Dokl. AN SSSR, 298:5 (1988), 1055–1060 | MR | Zbl