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Solvability of the dirichlet problem for a class of degenerate quasilinear elliptic equations

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Abstract

In a bounded domain of an n-dimensional space one considers the first boundary-value problem for second-order quasilinear elliptic equations having a divergent structure and admitting an implicit degeneracy of a definite type: viz., at the points where the solution vanishes the strong ellipticity of the equation is violated. The dependence of the principal part of the equation on the gradient of the solution is not assumed to be linear. One gives the definition of a generalized solution of the Dirichlet problem for such equations and one shows its existence under the condition of coerciveness (in a definite sense) and of pseudomonotonicity of the differential operator.

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Authors
  1. P. Z. Mkrtychyan
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 102, pp. 181–189, 1980.

The author is grateful to O. A. Ladyzhenskaya and to A. V. Ivanov for their constant interest and for repeated detailed discussions of this paper.

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Mkrtychyan, P.Z. Solvability of the dirichlet problem for a class of degenerate quasilinear elliptic equations. J Math Sci 22, 1274–1280 (1983). https://doi.org/10.1007/BF01460281

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  • DOI: https://doi.org/10.1007/BF01460281

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