Estimates of solutions to linear elliptic systems and equations

 Access to full text

 Full (PDF)

1. [1] H. Begehr, An approximation method for the Dirichlet problem of nonlinear elliptic systems in ${ℝ}^r$, Rev. Roumaine Math. Pures Appl. 27 (1982), 927-934. Zbl0497.35038
2. [2] H. Begehr, Boundary value problems for systems with Cauchy-Riemannian main part, in: Complex Analysis, Fifth Romanian-Finnish Seminar, Bucharest 1981, Lecture Notes in Math. 1014, Springer, Berlin 1983, 265-279. 
3. [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. [4] H. Begehr, Remark on Hilbert's boundary value problem for Beltrami systems, Proc. Roy. Soc. Edinburgh 98A (1984), 305-310. Zbl0562.35071
5. [5] H. Begehr and D. Q. Dai, Initial boundary value problem for nonlinear pseudoparabolic equations, Complex Variables Theory Appl., to appear. Zbl0814.35064
6. [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. Zbl0395.35047
7. [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. Zbl0424.35041
8. [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. Zbl0493.35045
9. [9] H. Begehr and R. P. Gilbert, Pseudohyperanalytic functions, Complex Variables Theory Appl. 9 (1988), 343-357. Zbl0652.30030
10. [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. Zbl0484.35074
11. [11] H. Begehr and G. N. Hile, Riemann boundary value problems for nonlinear elliptic systems, Complex Variables Theory Appl. 1 (1983), 239-261. Zbl0522.35075
12. [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. Zbl0674.35019
13. [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. Zbl0523.35086
14. [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. [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. Zbl0501.35070
16. [16] H. Begehr and G. C. Hsiao, The Hilbert boundary value problem for nonlinear elliptic systems, Proc. Roy. Soc. Edinburgh 94A (1983), 97-112. Zbl0532.35061
17. [17] H. Begehr and G. C. Hsiao, A priori estimates for elliptic systems, Z. Anal. Anwendungen 6 (1987), 1-21. Zbl0647.35069
18. [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. Zbl0682.35033
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. Zbl0701.35029
20. [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. Zbl0761.35070
21. [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. [22] L. Bers, Theory of Pseudoanalytic Functions, Courant Inst., New York 1953. 
23. [23] L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, in: Convegno Intern. Eq. Lin. Der. Parz., Trieste 1954, Edizioni Cremonese, Roma 1955, 111-140. 
24. [24] B. Bojarski, A boundary value problem for a system of elliptic first order partial differential equations, Dokl. Akad. Nauk SSSR 102 (1955), 201-204 (in Russian). 
25. [25] B. Bojarski, Homeomorphic solutions of Beltrami systems, ibid. 661-664 (in Russian). 
26. [26] B. Bojarski, On solutions of a linear elliptic system of differential equations in the plane, ibid. 871-874 (in Russian). 
27. [27] B. Bojarski, Some boundary value problems for elliptic systems with two independent variables, Ph.D. thesis, Moscow Univ., 1955 (in Russian). 
28. [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. [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. [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. Zbl0141.42902
31. [31] B. Bojarski, Theory of generalized analytic vectors, Ann. Polon. Math. 17 (1966), 281-320 (in Russian). 
32. [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. [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. [34] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman, London 1982. 
35. [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. [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. Zbl0729.35040
37. [37] A. Douglis, A function-theoretic approach to elliptic systems of equations in two variables, Comm. Pure Appl. Math. 6 (1953), 259-289. Zbl0050.31901
38. [38] A. Dzhuraev, Systems of Equations of Composite Type, Longman, Essex, 1989. 
39. [39] F. D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford 1966. Zbl0141.08001
40. [40] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin 1977. Zbl0361.35003
41. [41] R. P. Gilbert and J. Buchanan, First Order Elliptic Systems: A Function Theoretic Approach, Acad. Press, New York 1983. Zbl0529.35006
42. [42] R. P. Gilbert and G. N. Hile, Generalized hypercomplex function theory, Trans. Amer. Math. Soc. 195 (1974), 1-29. Zbl0298.30040
43. [43] R. P. Gilbert and G. N. Hile, Hypercomplex function theory in the sense of L. Bers, Math. Nachr. 72 (1976), 187-200. Zbl0341.30032
44. [44] W. Haak and W. Wendland, Lectures on Partial and Pfaffian Differential Equations, Pergamon Press, Oxford 1972. 
45. [45] P. Henrici, Applied and Computational Complex Analysis, Vol. 3, Wiley, New York 1986. 
46. [46] T. Iwaniec, Quasiconformal mapping problem for general nonlinear systems of partial differential equations, Symposia Math. 18 (1976), 501-517. 
47. [47] V. N. Monakhov, Boundary-value problems with free boundaries for elliptic systems of equations, Amer. Math. Soc., Providence, R.I., 1983. Zbl0532.35001
48. [48] N. I. Muskhelishvili, Singular Integral Equations, Noordhoff, Leyden 1977. 
49. [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. [50] W. Tutschke, Partielle Differentialgleichungen, Teubner, Leipzig 1983. 
51. [51] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford 1962. Zbl0127.03505
52. [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). Zbl0134.31201
53. [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. [54] V. S. Vinogradov, On some boundary value problems for quasilinear elliptic systems of first order in the plane, ibid., 579-581 (in Russian). Zbl0134.31203
55. [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). Zbl0246.35036
56. [56] G.-C. Wen and H. Begehr, Boundary Value Problems for Elliptic Equations and Systems, Longman, Essex 1990. Zbl0711.35038
57. [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. [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. [59] W. Wendland, Elliptic Systems in the Plane, Pitman, London 1979. Zbl0396.35001
60. [60] J. Wloka, Funktionenanalysis und Anwendungen, de Gruyter, Berlin 1971. 
61. [61] Z. Xu, Boundary value problems and function theory for spin-invariant differential operators, thesis, State University of Ghent, Gent 1989.