Abstract
One gives the formulation of the general boundary value problem for a large class of quasilinear equations of the divergence form, admitting a fixed ellipticity degeneracy on an arbitrary subset of the domain of variation of the independent variables. One finds conditions for the existence and the uniqueness of generalized solutions of the indicated problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
G. Fichera, “Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,” Atti Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. (8)5, No. 1, 1–30 (1956).
O. A. Oleinik and E. V. Radkevich, Second Order Equations with Nonnegative Characteristic Form, Amer. Math. Soc., Providence (1973).
E. V. Radkevich, “The second boundary value problem for second-order equations with a nonnegative characteristic form,” Vestn. Mosk. Univ., No. 4, 3–11 (1967).
J. J. Kohn and L. Nirenberg, “Degenerate elliptice parabolic equations of the second order,” Commun. Pure Appl. Math.,20, No. 4, 797–872 (1967).
R. S. Phillips and L. Sarason, “Ellipticoparabolic equations of the second order,” J. Math. Mech.,17, No. 9, 891–917 (1968).
A. V. Ivanov, “The first boundary value problem for quasilinear (A,\(\overrightarrow \beta\)) -elliptic equations,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,84, 45–88 (1979).
A. V. Ivanov, “The first and the second boundary value problems for quasilinear (A,\(\overrightarrow \beta\)) -elliptic equations,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,92, 171–181 (1979).
M. K. V. Murthy and G. Stampacchia, “Boundary value problems for some degenerate-elliptic operators,” Ann. Mat. Pura Appl.,80, 1–122 (1968).
S. N. Kruzhkov, “Boundary value problems for second-order degenerate elliptic equations,” Mat. Sb.,77, No. 3, 299–334 (1968).
N. S. Trudinger, “Linear elliptic operators with measurable coefficients,” Ann. Scuola Norm. Supp. Pisa,27, No. 2, 265–308 (1973).
A. V. Ivanov, “A boundary value problem for degenerate second-order linear parabolic equations,” Zap. Nauchn. Leningr. Otd. Mat. Inst.,14, 48–88 (1969).
A. V. Ivanov, “A boundary value problem for linear parabolic equations of the divergence type with measurable coefficients,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,52, 3–34 (1975).
A. V. Ivanov, “Properties of the solutions of nonstrictly and nonuniformly second-order parabolic linear and quasilinear equations with measurable coefficients,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,69, 45–64 (1977).
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).
P. Z. Mkrtychyan, “On the solvability of the first boundary value problem for higher order quasilinear elliptic equations which degenerate with respect to the independent variables,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.96, 187–204 (1980).
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence (1968).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 96, pp. 57–68, 1980.
Rights and permissions
About this article
Cite this article
Ivanov, A.V. General boundary value problem for quasilinear (A,\(\overrightarrow \beta\)) -elliptic equations. J Math Sci 21, 680–689 (1983). https://doi.org/10.1007/BF01094429
Issue Date:
DOI: https://doi.org/10.1007/BF01094429