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## Banach Center Publications

1992 | 27 | 1 | 45-63
Tytuł artykułu

### Estimates of solutions to linear elliptic systems and equations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Whenever nonlinear problems have to be solved through approximation methods by solving related linear problems a priori estimates are very useful. In the following this kind of estimates are presented for a variety of equations related to generalized first order Beltrami systems in the plane and for second order elliptic equations in $ℝ^m$. Different types of boundary value problems are considered. For Beltrami systems these are the Riemann-Hilbert, the Riemann and the Poincaré problem, while for elliptic equations the Dirichlet problem as well as entire solutions are involved.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
45-63
Opis fizyczny
Daty
wydano
1992
Twórcy
autor
• I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, D-1000 Berlin 33, Germany
Bibliografia
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• [3] H. Begehr, Boundary value problems for analytic and generalized analytic functions, in: Complex Analysis-Methods, Trends, and Applications, E. Lankau and W. Tutschke (eds.), Akad.- Verlag, Berlin 1983, 150-165.
• [4] H. Begehr, Remark on Hilbert's boundary value problem for Beltrami systems, Proc. Roy. Soc. Edinburgh 98A (1984), 305-310.
• [5] H. Begehr and D. Q. Dai, Initial boundary value problem for nonlinear pseudoparabolic equations, Complex Variables Theory Appl., to appear.
• [6] H. Begehr and R. P. Gilbert, Piecewise continuous solutions of pseudoparabolic equations in two space dimensions, Proc. Roy. Soc. Edinburgh 81A (1978), 153-173.
• [7] H. Begehr and R. P. Gilbert, On Riemann boundary value problems for certain linear elliptic systems in the plane, J. Differential Equations 32 (1979), 1-14.
• [8] H. Begehr and R. P. Gilbert, Boundary value problems associated with first order elliptic systems in the plane, in: Contemp. Math. 11, Amer. Math. Soc., 1982, 13-48.
• [9] H. Begehr and R. P. Gilbert, Pseudohyperanalytic functions, Complex Variables Theory Appl. 9 (1988), 343-357.
• [10] H. Begehr and G. N. Hile, Nonlinear Riemann boundary value problems for a semilinear elliptic system in the plane, Math. Z. 179 (1982), 241-261.
• [11] H. Begehr and G. N. Hile, Riemann boundary value problems for nonlinear elliptic systems, Complex Variables Theory Appl. 1 (1983), 239-261.
• [12] H. Begehr and G. N. Hile, Schauder estimates and existence theory for entire solutions of linear elliptic equations, Proc. Roy. Soc. Edinburgh 110A (1988), 101-123.
• [13] H. Begehr and G. C. Hsiao, Nonlinear boundary value problems for a class of elliptic systems, in: Komplexe Analysis und ihre Anwendungen auf partielle Differentialgleichungen, Martin-Luther Universität, Halle-Wittenberg. Wissensch. Beiträge 1980, 90-102.
• [14] H. Begehr and G. C. Hsiao, On nonlinear boundary value problems of elliptic systems in the plane, in: Ordinary and Partial Differential Equations, Proc. Dundee 1980, Lecture Notes in Math. 846, Springer, Berlin 1981, 55-63.
• [15] H. Begehr and G. C. Hsiao, Nonlinear boundary value problems of Riemann-Hilbert type, in: Contemp. Math. 11, Amer. Math. Soc., 1982, 139-153.
• [16] H. Begehr and G. C. Hsiao, The Hilbert boundary value problem for nonlinear elliptic systems, Proc. Roy. Soc. Edinburgh 94A (1983), 97-112.
• [17] H. Begehr and G. C. Hsiao, A priori estimates for elliptic systems, Z. Anal. Anwendungen 6 (1987), 1-21.
• [18] H. Begehr and G.-C. Wen, The discontinuous oblique derivative problem for nonlinear elliptic systems of first order, Rev. Roumaine Math. Pures Appl. 33 (1988), 7-19.
• [19] H. Begehr and G.-C. Wen, A priori estimates for the discontinuous oblique derivative problem for elliptic systems, Math. Nachr. 142 (1989), 307-336.
• [20] H. Begehr, G.-C. Wen and Z. Zhao, An initial and boundary value problem for nonlinear composite type systems of three equations, Math. Panonica 2 (1991), 49-61.
• [21] H. Begehr and Z. Y. Xu, Nonlinear half-Dirichlet problems for first order elliptic equations in the unit ball of $ℝ^m$ (m ≥ 3), Appl. Anal., to appear.
• [22] L. Bers, Theory of Pseudoanalytic Functions, Courant Inst., New York 1953.
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• [27] B. Bojarski, Some boundary value problems for elliptic systems with two independent variables, Ph.D. thesis, Moscow Univ., 1955 (in Russian).
• [28] B. Bojarski, Generalized solutions of a system of differential equations of the first order of elliptic type with discontinuous coefficients, Mat. Sb. 43 (85) (1957), 451-503 (in Russian).
• [29] B. Bojarski, Studies on elliptic equations in plane domains and boundary value problems of function theory, Habilitation thesis, Steklov Inst., Moscow 1960, 320 pp. (in Russian).
• [30] B. Bojarski, Subsonic flow of compressible fluid, Arch. Mech. Stos. 18 (1966), 497-519; also in: The Math. Problems in Fluid Mechanics, Polish Acad. Sci., Warszawa 1967, 9-32.
• [31] B. Bojarski, Theory of generalized analytic vectors, Ann. Polon. Math. 17 (1966), 281-320 (in Russian).
• [32] B. Bojarski, Quasiconformal mappings and general structural properties of systems of non-linear equations elliptic in the sense of Lavrentiev, Symposia Math. 18 (1976), 485-499.
• [33] B. Bojarski and I. Iwaniec, Quasiconformal mappings and non-linear elliptic equations in two variables. I, II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 473-478; 479-484.
• [34] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman, London 1982.
• [35] D. Q. Dai, On an initial boundary value problem for nonlinear pseudoparabolic equations with two space variables, Complex Variables Theory Appl., to appear.
• [36] R. Delanghe, F. Sommen and Z. Y. Xu, Half-Dirichlet problems for powers of the Dirac operator in the unit ball of $ℝ^m$ (m ≥ 3), Bull. Soc. Math. Belg. Sér. B 42 (1990), 409-429.
• [37] A. Douglis, A function-theoretic approach to elliptic systems of equations in two variables, Comm. Pure Appl. Math. 6 (1953), 259-289.
• [38] A. Dzhuraev, Systems of Equations of Composite Type, Longman, Essex, 1989.
• [39] F. D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford 1966.
• [40] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin 1977.
• [41] R. P. Gilbert and J. Buchanan, First Order Elliptic Systems: A Function Theoretic Approach, Acad. Press, New York 1983.
• [42] R. P. Gilbert and G. N. Hile, Generalized hypercomplex function theory, Trans. Amer. Math. Soc. 195 (1974), 1-29.
• [43] R. P. Gilbert and G. N. Hile, Hypercomplex function theory in the sense of L. Bers, Math. Nachr. 72 (1976), 187-200.
• [44] W. Haak and W. Wendland, Lectures on Partial and Pfaffian Differential Equations, Pergamon Press, Oxford 1972.
• [45] P. Henrici, Applied and Computational Complex Analysis, Vol. 3, Wiley, New York 1986.
• [46] T. Iwaniec, Quasiconformal mapping problem for general nonlinear systems of partial differential equations, Symposia Math. 18 (1976), 501-517.
• [47] V. N. Monakhov, Boundary-value problems with free boundaries for elliptic systems of equations, Amer. Math. Soc., Providence, R.I., 1983.
• [48] N. I. Muskhelishvili, Singular Integral Equations, Noordhoff, Leyden 1977.
• [49] W. Tutschke, Reduction of the problem of linear conjugation for first order nonlinear elliptic systems in the plane to an analogous problem for holomorphic functions, in: Lecture Notes in Math. 798, Springer, Berlin 1980, 446-455.
• [50] W. Tutschke, Partielle Differentialgleichungen, Teubner, Leipzig 1983.
• [51] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford 1962.
• [52] V. S. Vinogradov, On a boundary value problem for linear elliptic systems of first order in the plane, Dokl. Akad. Nauk SSSR 118 (1958), 1059-1062 (in Russian).
• [53] V. S. Vinogradov, On the boundedness of solutions of boundary value problems for linear elliptic systems of first order in the plane, ibid. 121 (1958), 399-402 (in Russian).
• [54] V. S. Vinogradov, On some boundary value problems for quasilinear elliptic systems of first order in the plane, ibid., 579-581 (in Russian).
• [55] V. S. Vinogradov, A certain boundary value problem for an elliptic system of special form, Differentsial'nye Uravneniya 7 (1971), 1226-1234, 1341 (in Russian).
• [56] G.-C. Wen and H. Begehr, Boundary Value Problems for Elliptic Equations and Systems, Longman, Essex 1990.
• [57] W. Wendland, An integral equation method for generalized analytic functions, in: Constructive and Computational Methods for Differential and Integral Equations, Lecture Notes in Math. 430, Springer, Berlin 1974, 414-452.
• [58] W. Wendland, On the imbedding method for semilinear first order elliptic systems and related finite element methods, in: Continuation Methods, Acad. Press, New York 1978, 277-336.
• [59] W. Wendland, Elliptic Systems in the Plane, Pitman, London 1979.
• [60] J. Wloka, Funktionenanalysis und Anwendungen, de Gruyter, Berlin 1971.
• [61] Z. Xu, Boundary value problems and function theory for spin-invariant differential operators, thesis, State University of Ghent, Gent 1989.
Typ dokumentu
Bibliografia
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