Towards the Cauchy problem for the Laplace equation

 Access to full text

 Full (PDF)

1. [1] N. A. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. 36 (1957), 235-249. Zbl0084.30402
2. [2] M. A. Atakhodzhaev and U. E. Kobilov, The Cauchy problem for the Laplace equation in an infinite three-dimensional layer, in: Direct and Inverse Problems for Partial Differential Equations and their Applications 186, Fan, Tashkent 1978, 62-75 (in Russian). 
3. [3] J. Baumeister, Stable Solution of Inverse Problems, Vieweg & Sohn, Braunschweig 1987. 
4. [4] P. S. Bondarenko and A. V. Rilov, The estimation of the modulus of continuity of the inverse Laplace transform in the solution of the Cauchy problem for Laplace's equation, in: Approximate and Qualitative Methods of the Theory of Differential Equations, Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev 1971, 167-174 (in Russian). 
5. [5] A. P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. Zbl0080.30302
6. [6] J. R. Cannon and P. DuChateau, Approximating the solution to the Cauchy problem for Laplace's equation, SIAM J. Numer. Anal. 14 (1977), 473-483. Zbl0366.65053
7. [7] J. R. Cannon and J. Douglas, Jr., The approximation of harmonic and parabolic functions on half-spaces from interior data, in: Numerical Analysis of Partial Differential Equations (C.I.M.E. $2^0$ Ciclo, Ispra 1967), Edizioni Cremonese, Rome 1968, 193-230. 
8. [8] J. R. Cannon and K. Miller, Some problems in numerical analytic continuation, SIAM J. Numer. Anal. 2 (1965), 87-98. Zbl0214.14805
9. [9] T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys. B 26 (1939), 1-9. Zbl0022.34201
10. [10] L. A. Chudov, Difference methods for solving Cauchy problem for Laplace's equation, Dokl. Akad. Nauk SSSR 143 (1962), 789-801; English transl.: Soviet Math. Dokl. 3 (1962), 499-503. 
11. [11] D. L. Colton, Improperly posed initial value problem for self-adjoint hyperbolic and elliptic equations, SIAM J. Math. Anal. 4 (1973), 42-51. Zbl0224.35005
12. [12] D. L. Colton, Partial Differential Equations in the Complex Domain, Pitman, London 1976. Zbl0323.35003
13. [13] J. Conlan and R. P. Gilbert, Non-linear initial data for second and higher order semi-linear elliptic equations, J. Reine Angew. Math. 276 (1975), 1-14. Zbl0311.35045
14. [14] J. Conlan and G. N. Trytten, Pointwise bounds in the Cauchy problem for elliptic systems of partial differential equations, Arch. Rational Mech. Anal. 22 (1966), 143-152. Zbl0166.37601
15. [15] H. O. Cordes, Über die Bestimmheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. Wiss. Göttingen Math.-Phys. IIa 11 (1956), 239-258. Zbl0074.08002
16. [16] Dinh Nho Hào and R. Gorenflo, A noncharacteristic Cauchy problem for the heat equation, Acta Appl. Math. 24 (1991), 1-27. (See also: Dinh Nho Hào and R. Gorenflo, An ill-posed problem for the heat equation, Z. Angew. Math. Mech. 71 (1991), T763-T766). Zbl0741.35090
17. [17] A. Douglis, Uniqueness in Cauchy problems for elliptic systems of equations, Comm. Pure Appl. Math. 13 (1960), 593-608. Zbl0095.30601
18. [18] Yu. A. Dubinskiĭ, The algebra of pseudodifferential operators with analytic symbols and its applications to mathematical physics, Uspekhi Mat. Nauk 37 (1982), 97-137; English transl.: Russian Math. Surveys 37 (1982), 107-153. 
19. [19] G. A. Dzhafarli, A uniqueness theorem for the solutions of an elliptic system in the half-plane, Dokl. Akad. Nauk Azerbaidzhan. SSR 26 (1970), 12-14 (in Russian). Zbl0213.37603
20. [20] Kh. Sh. Dzhuraev, On a solution of the Cauchy problem for the Laplace equation, Dokl. Akad. Nauk Tadzhik. SSR 29 (1986), 506-509 (in Russian). 
21. [21] R. S. Falk and P. B. Monk, Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for the Poisson's equation, Math. Comp. 47 (1986), 135-149. Zbl0623.65095
22. [22] Yu. L. Gapanenko, Stability of the solution of the Cauchy problem for the Laplace equation on a weak compactum, in: Numerical Mathematics and Sofware, Moskov. Gos. Univ., 1985, 44-50 (in Russian). 
23. [23] V. I. Gorbaĭchuk, Estimates of the accuracy of an approximate solution for the Cauchy problem for the Laplace equation in an infinite strip, Ukrain. Mat. Zh. 32 (1980), 731-736; English transl.: Ukrainian Math. J. 32 (1980), 489-494. 
24. [24] R. Gorenflo, Funktionentheoretische Bestimmung des Aussenfeldes zu einer zweidimensionalen magnetohydrostatischen Konfiguration, Z. Angew. Math. Phys. 16 (1965), 279-290. Zbl0127.16903
25. [25] R. Gorenflo, Behandlung ebener magnetohydrostatischer Gleichgewichtsprobleme mittels komplexer Analysis, in: Tagung Freie Randwertaufgaben, E. Grafarend und N. Weck (eds.), Mitteilungen aus dem Institut für Theoretische Geodäsie der Universität Bonn, N.4, 1972, 2-18. Zbl0341.76064
26. [26] J. Hadamard, Sur les problèmes aux derivées partielles et leur signification physique, Bull. Univ. Princeton 13 (1902), 49-32. 
27. [27] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale Univ. Press, 1923. Zbl49.0725.04
28. [28] H. D. Han, The finite element method in the family of improperly posed problems, Math. Comp. 38 (1982), 55-65. Zbl0476.65064
29. [29] E. Heinz, Über die Eindeutigkeit beim Cauchyschen Anfangswertproblem einer elliptischen Differentialgleichung zweiter Ordnung, Nachr. Akad. Wiss. Göttingen Math.-Phys. IIa 1 (1955), 1-12. Zbl0067.07503
30. [30] L. Hörmander, Linear Partial Differential Operators, Springer, Berlin 1976. Zbl0321.35001
31. [31] L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21-64. Zbl0546.35023
32. [32] V. K. Ivanov, The Cauchy problem for the Laplace equation in an infinite strip, Differentsial'nye Uravneniya 1 (1965), 131-136 (in Russian). 
33. [33] F. John, A note on ``improper'' problems in partial differential equations, Comm. Pure Appl. Math. 8 (1955), 494-495. 
34. [34] F. John, Differential Equations with Approximate and Improper Data, New York University, 1955. 
35. [35] F. John, Continuous dependence on the data for solutions of partial differential equations with a predescribed bound, Comm. Pure Appl. 13 (1960), 551-586. 
36. [36] L. A. Knizhnerman, Numerical solution of the Cauchy problem for the Laplace equation by means of an expansion in Fourier-Chebyshev series, Izv. Akad. Nauk SSSR Ser. Fiz. Zemli 1984 (10), 76-81 (in Russian). 
37. [37] R. J. Knops and L. E. Payne, Uniqueness and continuous dependence of the null solution in the Cauchy problem for a nonlinear elliptic system, in: Inverse and Improperly Posed Problems in Differential Equations, G. Anger (ed.), Akademie-Verlag, Berlin 1979, 151-160. 
38. [38] T. I. Korolyuk, The Cauchy problem for the Laplace equation, Izv. Vyssh. Uchebn. Zaved. Mat. 130 (1973), 53-55 (in Russian). 
39. [39] H. Kumano-, go, On the uniqueness of the solution of the Cauchy problem and the unique continuation theorem for elliptic equations, Osaka Math. J. 14 (1962), 182-212. 
40. [40] E. M. Landis, Certain properties of equations of elliptic type, Dokl. Akad. Nauk SSSR 107 (1956), 640-643 (in Russian). Zbl0075.28201
41. [41] E. M. Landis, Some questions in the qualitative theory of elliptic and parabolic equations, Uspekhi Mat. Nauk 14 (1) (1959), 21-85; English transl. in Amer. Math. Soc. Transl. (2) 20 (1962), 173-238. 
42. [42] E. M. Landis, Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables), Uspekhi Mat. Nauk 18 (1) (1963), 3-62; English transl. in Russian Math. Surveys 18 (1963), 1-62. Zbl0125.05802
43. [43] M. M. Lavrent'ev, On the Cauchy problem for the Laplace equation, Izv. Akad. Nauk SSSR Ser. Mat. 120 (1956), 819-842 (in Russian). 
44. [44] M. M. Lavrent'ev, On the Cauchy problem for linear elliptic equations of second order, Dokl. Akad. Nauk SSSR 112 (1957), 195-197 (in Russian). 
45. [45] M. M. Lavrent'ev, Some Improperly Posed Problems in Mathematical Physics, Springer, New York 1967. 
46. [46] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishat-, skiĭ, Ill-posed Problems of Mathematical Physics and Analysis, Transl. Math. Monographs, Amer. Math. Soc., Providence, R.I., 1986. 
47. [47] N. Lerner, Unicité du problème de Cauchy pour des opérateurs elliptiques, Ann. Sci. École Norm. Sup. (4) 17 (1984), 469-505. Zbl0562.35006
48. [48] H. A. Levine and S. Vessella, Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type, Rend. Circ. Mat. Palermo 123 (1986), 161-183. 
49. [49] B. H. Li and Y. Q. Li, On the initial value problem of the Laplace equation, J. Systems Sci. Math. Sci. 7 (1987), 1-6. 
50. [50] O. A. Liskovets, A solution of the Cauchy problem for the Laplace equation by a generalized method of the summability of series, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1970 (4), 68-74 (in Russian). 
51. [51] V. G. Mazya and V. P. Khavin, On the solutions of the Cauchy problem for Laplace's equation (solvability, normality, approximation), Trans. Moscow Math. Soc. 30 (1974), 65-117. 
52. [52] L. A. Medeiros, Remarks on a non well-posed problem, Proc. Roy. Soc. Edinburgh 102A (1986), 131-140. Zbl0602.35068
53. [53] S. M. Mergelyan, Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation, Uspekhi Mat. Nauk 11 (1956), 3-26 (in Russian). 
54. [54] F. Meyer und H. U. Schmidt, Torusartige Plasmakonfigurationen ohne Gesamtstrom durch ihren Querschnitt im Gleichgewicht mit einem Magnetfeld, Z. Naturforsch. 13a (1958), 1005-1015. Zbl0084.23401
55. [55] K. Miller, Three circle theorems in partial differential equations and applications to improperly posed problems, Arch. Rational Mech. Anal. 16 (1964), 126-154. Zbl0145.14203
56. [56] K. Miller, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal. 1 (1970), 52-74. Zbl0214.14804
57. [57] S. Mizohata, Unicité dans les problèmes de Cauchy pour quelques équations différentielles elliptiques, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 31 (1958), 121-128. Zbl0090.31402
58. [58] S. Mizohata, Unicité du prolongement des solutions des équations elliptiques du quatrième ordre, Proc. Japan Acad. 34 (1958), 687-692. Zbl0085.08501
59. [59] C. Müller, On the behavior of the solutions of the differential equation Δu = F(x,u) in the neighborhood of a point, Comm. Pure Appl. Math. 7 (1954), 505-514. 
60. [60] S. M. Nikol'skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin 1975 (translated from the Russian). 
61. [61] L. Nirenberg, Uniqueness in Cauchy problems for differential equations with constant leading coefficients, Comm. Pure Appl. Math. 10 (1957), 89-105. Zbl0077.09402
62. [62] L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal. 5 (1960), 35-45. Zbl0094.29801
63. [63] L. E. Payne, On a priori bounds in the Cauchy problem for elliptic equations, SIAM J. Math. Anal. 1 (1970), 82-89. Zbl0199.16603
64. [64] L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, Penn., 1975. Zbl0302.35003
65. [65] L. E. Payne and D. Sather, On some improperly posed problems for quasilinear equations of mixed type, Trans. Amer. Math. Soc. 128 (1967), 135-141. Zbl0154.11302
66. [66] L. E. Payne and D. Sather, On some improperly posed problems for the Chaplygin equation, J. Math. Anal. Appl. 19 (1967), 67-77. Zbl0149.30803
67. [67] L. E. Payne and D. Sather, On an initial-boundary value problem for a class of degenerate elliptic operators, Ann. Mat. Pura Appl. 78 (1968), 323-337. Zbl0167.39902
68. [68] A. P. Poddubnyak and V. E. Emets, The Cauchy problem for the Laplace equation in an infinite n + 1-dimensional layer, Mat. Metody i Fiz.-Mekh. Polya 15 (1982), 13-15 (in Russian). Zbl0482.35033
69. [69] M. H. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc. 95 (1960), 81-91. Zbl0094.07901
70. [70] C. Pucci, Sui problemi di Cauchy non ``ben posti'', Rend. Accad. Naz. Lincei 8 (18) (1955), 473-477. Zbl0065.07805
71. [71] C. Pucci, Discussione del problema di Cauchy per le equazioni di tipo ellittico, Ann. Mat. Pura Appl. 46 (1958), 131-153. Zbl0092.10001
72. [72] C. Pucci, Some Topics in Parabolic and Elliptic Equations, Lecture Series No. 36, Institute for Fluid Mechanics and Applied Mathematics, University of Maryland, 1958. 
73. [73] A. Romanovich and A. V. Fursikov, On the question of the solvability of the Cauchy problem for the Laplace operator, Vestnik Moskov. Gos. Univ. Ser. I Mat. Mekh. 1987 (3), 78-80. 
74. [74] J.-C. Saut et B. Scheurer, Sur l'unicité du problème de Cauchy et le prolongement unique pour des équations elliptiques à coefficients non localement bornés, J. Differential Equations 43 (1982), 28-43. Zbl0431.35017
75. [75] J.-C. Saut et B. Scheurer, Unique combination and uniqueness of the Cauchy problem for elliptic equations with unbounded coefficients, in: Nonlinear Partial Differential Equations and their Applications, Collège de France Séminaire, Vol. V (Paris 1981/ 1982), Res. Notes in Math. 93, Pitman, Boston, Mass., 1983, 260-275. 
76. [76] P. W. Schaefer, On the Cauchy problem for an elliptic system, Arch. Rational Mech. Anal. 20 (1965), 391-412. Zbl0144.15101
77. [77] P. W. Schaefer, Pointwise bounds in the Cauchy problem for an elliptic system, SIAM J. Appl. Math. 15 (1967), 665-677. Zbl0152.11103
78. [78] P. W. Schaefer, Improvable estimates in some non-well-posed problems for a system of elliptic equations, SIAM J. Math. Anal. 4 (1973), 447-455. Zbl0256.35033
79. [79] P. W. Schaefer, On uniqueness, stability, and pointwise estimates in the Cauchy problem for coupled elliptic equations, Quart. Appl. Math. 31 (1973), 321-328. Zbl0268.35040
80. [80] S. D. Shalaginov, The Cauchy problem for the Laplace equation in a complex space, Differentsial'nye Uravneniya 16 (1980), 947-949 (in Russian). Zbl0436.35025
81. [81] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems, Wiley, New York 1977 (Russian third ed., 1986). 
82. [82] N. V. Tkachenko, Solution of a certain Cauchy problem for the Laplace equation, in: Math. Physics, Leningrad Gos. Ped. Inst., Leningrad 1987, 27-32 (in Russian). 
83. [83] G. Trytten, Pointwise bounds for solutions of the Cauchy problem for elliptic equations, Arch. Rational Mech. Anal. 13 (1963), 222-244. Zbl0115.08602
84. [84] Tran Duc Van, On the pseudodifferential operators with real analytic symbols and their applications, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), 803-825. Zbl0732.35117
85. [85] Tran Duc Van, Dinh Nho Hào and R. Gorenflo, Approximating the solution to the Cauchy problem and the boundary problem for the Laplace equation, to appear. Zbl0820.35032
86. [86] Tran Duc Van, Dinh Nho Hào, Trinh Ngoc Minh and R. Gorenflo, On the Cauchy problems for systems of partial differential equations with a distinguished variable, Numer. Funct. Anal. Optim. 12 (1&2) (1991), 213-236. Zbl0789.35074
87. [87] Tran Duc Van, Nguyen Duy Thai Son and Dinh Zung, Approximately solving Cauchy problem for the wave equation by the method of differential operators of infinite order, Acta Math. Vietnam. 13 (2) (1988), 127-136. Zbl0683.65082
88. [88] Trinh Ngoc Minh and Tran Duc Van, Cauchy problems for systems of partial differential equations with a distinguished variable, Dokl. Akad. Nauk SSSR 284 (1985), 507-510; English transl.: Soviet Math. Dokl. 32 (1985), 562-565. 
89. [89] Trinh Ngoc Minh, Linear differential operators of infinite order and their applications, Acta Math. Vietnam. 12 (1) (1987) 101-124. 
90. [90] P. N. Vabishchevich, On the solution of the Cauchy problem for the Laplace equations in a doubly connected domain, Dokl. Akad. Nauk SSSR 241 (1978), 1257-1260; English transl.: Soviet Math. Dokl. 19 (1978), 976-980. 
91. [91] P. N. Vabishchevich, Numerical solution of the Cauchy problem for elliptic equations and systems, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 1979 (3), 3-10 (in Russian). Zbl0439.65082
92. [92] P. N. Vabishchevich, V. B. Glasko and Yu. A. Kriksin, Solution of the Hadamard problem by a Tikhonov-regularizing algorithm, U.S.S.R. Comput. Math. and Math. Phys. 19 (1979), 103-112. Zbl0462.35087
93. [93] P. N. Vabishchevich and P. A. Pulatov, A method of numerical solution of the Cauchy problem for elliptic equations, Vestnik Moskov. Gos. Univ. Ser. XV Vychisl. Mat. Kibernet. 1984, 3-8. Zbl0553.65061
94. [94] K. Watanabe, On the uniqueness of the Cauchy problem for certain elliptic equations with triple characteristics, Tôhoku Math. J. (2) 23 (1971), 473-490. Zbl0237.35032
95. [95] K. Watanabe, A unique continuation theorem for an elliptic operator of two independent variables with nonsmooth double characteristics, Osaka J. Math. 10 (1973), 243-246. Zbl0279.35027
96. [96] K. Watanabe, Remarque sur l'unicité dans les problèmes de Cauchy pour les opérateurs différentiels elliptiques à caractéristiques de multiplicité constante et au plus triple, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), A1385-A1386. Zbl0355.35027
97. [97] K. Watanabe et C. Zuily, Unicité du problème de Cauchy pour une classe d'opérateurs différentiels elliptiques à caractéristiques de multiplicité variable, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), A627-A630. Zbl0337.35016
98. [98] K. Watanabe et C. Zuily, Unicité du problème de Cauchy pour une classe d'opérateurs différentiels elliptiques à caractéristiques de multiplicité variable, Sém. Goulaouic-Schwartz 1976/1977, Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 15, 9 pp., Centre Math., École Polytech., Palaiseau 1977. Zbl0337.35016
99. [99] K. Watanabe et C. Zuily, On the uniqueness of the Cauchy problem for elliptic differential operators with smooth charateristics of variable multiplicity, Comm. Partial Differential Equations 2 (1977), 831-855. Zbl0403.35015
100. [100] A. Yanushauskas, A Cauchy problem for Laplace's equation and the multiplication operation for harmonic functions, Dokl. Akad. Nauk SSSR 159 (1964), 286-289 (in Russian). 
101. [101] A. Yanushauskas, Entire solutions of second order elliptic equations, Siberian Math. J. 11 (1970), 863-869. Zbl0213.37601
102. [102] A. Yanushauskas, On the Cauchy problem for the Laplace equation with three independent variables, ibid. 16 (1975), 1040-1047. 
103. [103] Sh. Yarmukhamedov, On the Cauchy problem for Laplace's equation, Dokl. Akad. Nauk SSSR 235 (1977), 281-283, English transl.: Soviet Math. Dokl. 18 (1977), 939-882. Zbl0387.35022
104. [104] Sh. Yarmukhamedov, The Cauchy problem for the Laplace equation in an infinite domain, Dokl. Akad. Nauk UzSSR 1980 (12), 9-10 (in Russian). Zbl0499.35037
105. [105] Sh. Yarmukhamedov, The Cauchy problem for the Laplace equation in M. M. Lavrent'ev's formulation, in: Ill-posed Problems of Mathematical Physics and Analysis, A. S. Alekseev (ed.), Nauka, Sibirsk. Otdel., Novosibirsk 1984, 203-209 (in Russian). 
106. [106] C. Zuily, Unicité du problème de Cauchy pour des opérateurs elliptiques à caractéristiques de hautes multiplicités, Comm. Partial Differential Equations 10 (1985), 219-244. Zbl0576.35039
107. 1. Dinh Nho Hào, A mollification method for ill-posed problems, preprint A-92-35, FB Mathematik, FU Berlin. 
108. 2. A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation, Trans. Moscow Math. Soc. 52 (1990), 139–176. Zbl0716.35023
109. 3. M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace’s equation, SIAM J. Appl. Math. 5 (1991), 1653–1675. Zbl0769.35005