For a class of quasilinear elliptic equations of an arbitrary order, admitting a fixed weak degeneracy with respect to the independent variables, one proves an existence theorem for the generalized solution of the Dirichlet problem. On gives conditions under which such a solution is unique.
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A. V. Ivanov and P. Z. Mkrtychyan, “On the solvability of the first boundary value problem for certain classes of degenerate quasilinear elliptic equations of the second order,” Tr. Mat. Inst. Akad. Nauk SSSR,147, 14–39 (1980).
Ven'-Tuan Lu, “On imbedding theorems for spaces of functions with partial derivatives, summable with various powers,” Vestn. Leningr. Univ., No. 7, 2, 23–37 (1961).
S. N. Kruzhkov, “Boundary value problems for second-order degenerate elliptic equations,” Mat. Sb.,77, No. 3, 299–334 (1968).
M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York (1964).
F. E. Browder, “Nonlinear elliptic boundary-value problems,” Bull. Am. Math. Soc.,69, No. 6, 862–874 (1963).
G. J. Minty, “On a monotonicity method for the solution of nonlinear equations in Banach spaces,” Proc. Nat. Acad. Sci. USA,50, No. 6, 1038–1041 (1963).
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).
A. V. Ivanov, “The first boundary value problem for quasilinear (A,\(\overline \beta\)) -elliptic equations,” Zap. Nachn. Sem. Leningr. Otd. Mat. Inst.84, 45–88 (1979).
Yu. A. Dubinskii, Nonlinear elliptic and parabolic equations, in: Itogi Nauki Tekh., Sovrem. Probl. Mat., Vol. 9, Moscow (1976).
J. L. Lions, Quelques méthodes de resolution des problèmes aux limites nonlinéaires, Dunod, Gauthier-Villars, Paris (1969).
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 187–204, 1980.
The author expresses his gratitude to A. V. Ivanov for his constant interest in the paper and for repeated discussions.
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Mkrtychyan, P.Z. Solvability of the first boundary value problem for higher order quasilinear elliptic equations which degenerate with respect to the independent variables. J Math Sci 21, 783–797 (1983). https://doi.org/10.1007/BF01094441
- Generalize Solution
- Elliptic Equation
- Dirichlet Problem
- Existence Theorem
- Arbitrary Order