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Direct and Inverse Problems in Potential Theory

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Nonlinear Evolution Equations and Potential Theory

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Abstract

The aim of this paper is to sketch the most important direct problems (boundary value problems and initial value problems) of linear elliptic, parabolic and hyperbolic differential equations and some inverse problems corresponding to these equations. Both types of problems are divided into two classes. The first one is the class of properly posed problems, the other is the class of improperly posed problems. The Dirichlet problem for elliptic equations and parabolic equations and the Cauchy problem for hyperbolic equations are properly posed problems, the Dirichlet problem for hyperbolic equations and the inverse problem for the Laplace equation and the heat equation are improperly posed problems. There exist also inverse (improperly posed) problems concerning hyperbolic equations.

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References

  1. S. Agmon, Maximum properties for solutions of higher order elliptic equations. Bull. Amer. math. Soc. 66, 77–80 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. II. Comm. pure appl. Math. 12, 623–727 (1959), 17, 35–92 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Albinus, Bemerkungen zu einer potentialtheoretischen Methode. Math. Nachr. 55, 381–397 (1972).

    Article  MathSciNet  Google Scholar 

  4. G. Albinus, Über das im Sinne von N. Wiener verallgemeinerte Dirichletsche Problem für die Helmholtzsche Schwingungsgleichung in beliebigen beschränkten Gebieten (to appear in Math. Nachr.).

    Google Scholar 

  5. G. Albinus, N. Boboc und P. Mustata, A priori evaluations for the Dirichlet problem associated with a linear elliptic operator (to appear in Math. Nachr.).

    Google Scholar 

  6. G. Anger, Stetige Potentiale und deren Verwendung für einen Neuaufbau der Potentialtheorie. Dissertation, Technische Hochschule Dresden, 1957.

    Google Scholar 

  7. G. Anger, Sur le rôle des potentiels continues dans les fondements de la théorie du potentiel. Séminaire de théorie du potentiel, 2eme année, 1957/58, Institut H. Poincaré, Université de Paris.

    Google Scholar 

  8. G. Anger, Balayage-Prinzip und stetige Projektionen. Wiss. Z. Techn. Universität Dresden 11, 417–426 (1962).

    Google Scholar 

  9. G. Anger, Funktionalanalytische Betrachtungen bei Differentialgleichungen unter Verwendung von Methoden der Potentialtheorie, I. Akademie-Verlag, Berlin 1967.

    MATH  Google Scholar 

  10. G. Anger, Eine allgemeine Kerntheorie, I. Sp-Kerne. Math. Nachr. 37, 153–176 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Anger, Potentialtheoretische Untersuchungen über Randwertaufgaben, deren Randoperatoren Ableitungen beliebiger Ordnung enthalten. Kolloquiumsbericht “Elliptische Differentialgleichungen, II.”, Berlin 1971, S. 15–78.

    Google Scholar 

  12. G. Anger, Eindeutigkeitssätze und Approximationssätze für Potentiale, I. Math. Nachr. 43, 123–142 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Anger, Eindeutigkeitssätze und Approximationssätze für Potentiale, II. Math. Nachr. 50, 229–244 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  14. Google Scholar 

  15. G. Anger, B.-W. Schulze and G. Wildenhain, Neuere Entwicklungen auf dem Gebiet der Potentialtheorie (to appear in “Die Entwicklung der Mathematik in der DDR”, Berlin 1974).

    Google Scholar 

  16. G. Anger and B.-W. Schulze, Some remarks on harmonic measures and inverse problems (to appear in “Beiträge zur Analysis und Angewandten Mathematik”, Berlin 1974).

    Google Scholar 

  17. Google Scholar 

  18. H. Bauer, Harmonische Räume und ihre Potentialtheorie. Berlin-Heidelberg- New York 1966.

    MATH  Google Scholar 

  19. N. Boboc et P. Mustatâ, Espaces harmoniques associés aux opérateurs différentiels linéaires du second ordre de type elliptique. Berlin-Heidelberg-New York 1968.

    MATH  Google Scholar 

  20. M. Brelot, Eléments de la théorie classique du potentiel. Centre de Documentation Univ., 4e édit., Paris 1969.

    Google Scholar 

  21. M. Brelog (editor), Potential theory. Centro Internazionale Matematico Estivo. Edizioni Cremonese, Roma 1970.

    Google Scholar 

  22. M. Brelot, Les étapes et les aspects multiples de la théorie du potentiel. Enseignement math., II. Sér. 18, 1, 1–36 (1972).

    MathSciNet  MATH  Google Scholar 

  23. F.E. Browder, Functional analysis and partial differential equations, II. Math. Ann. 145, 81–22 6 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Canfora, Teorema del massimo modulo e teorema di esistenza per il problema di Dirichlet relativo di sistemi fortemente ellitici. Ricerche Mat. 15, 249–294 (1966).

    MathSciNet  MATH  Google Scholar 

  25. 23.G. Choquet, Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953/54).

    Article  MathSciNet  Google Scholar 

  26. G. Choquet, Lectures on analysis II, New York-Amsterdam, 1969.

    MATH  Google Scholar 

  27. C. Constantinescu and A. Cornea, Potential theory and harmonic spaces. Berlin-Heidelberg-New York 1972.

    Google Scholar 

  28. Google Scholar 

  29. Google Scholar 

  30. Google Scholar 

  31. M. M. Lavrentiev, Some improperly posed problems of mathematical physics. Berlin-Heidelberg-New York 1967.

    MATH  Google Scholar 

  32. M.M. Lavrentiev, V.O. Romanov and V.G. Vasiliev, Multidimensional inverse problems for differential equations. Berlin-Heidelberg-New York 1970.

    MATH  Google Scholar 

  33. J. Lukeš, A new type of generalized solution of the Dirichlet problem for the heat equation. Nonlinear Evolution Equations and Potential Theory, Academia Praha 1974.

    Google Scholar 

  34. Google Scholar 

  35. S.G. Michlin, Lehrgang der mathematischen Physik. Berlin 1972 (Übers, a.d.Russischen).

    MATH  Google Scholar 

  36. C. Miranda, Teorema del massimo modulo e teorema di esistenza e di uniceta per il problema Dirichlet relativo alle equazioni ellitiche in due variabili. Ann. Mat. pura appl., IV. Ser., 46, 265–311 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  37. A.I. Prilepko, über Existenz und Eindeutigkeit von Lösungen inverser Probleme der Potentialtheorie (to appear in Math. Nachr.).

    Google Scholar 

  38. Google Scholar 

  39. M. Schechter, On LP estimates and regularity I, II, III. Amer. J. Math. 85, 1–13 (1963), Math. Scandinav. 13, 47–69 (1963), Ricerche Mat. 13, 192–206 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  40. B.-W. Schulze, Potentiale bei der Wellengleichung im R2 und Charakterisierung der Mengen der Kapazität Null. Kolloquium “Elliptische Differentialgleichungen I”, Berlin 1970, S. 137–157.

    Google Scholar 

  41. B.-W. Schulze, Mengen der Kapazität Null für nichtelliptische Differentialgleichungen. Das Dirichlet-Problem für uxy = 0. Kolloquium “Elliptische Differentialgleichungen II”, Berlin 1971, S. 217–246.

    Google Scholar 

  42. B.-W. Schulze, Nullmengensysteme in der Potentialtheorie. Math. Nachr. 49, 293–309 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  43. B.-W. Schulze, Ein Balayage-Prinzip für korrekte Randwert-Probleme I. Math. Nachr. 56, 307–344 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  44. B.-W. Schulze, Ein Balayage-Prinzip für korrekte Randwert-Probleme II. (to appear in Math. Nachr.).

    Google Scholar 

  45. Google Scholar 

  46. B.-W. Schulze, Anwendungen des Balayage-Prinzipes auf allgemeine Randwert-Probleme für Systeme partieller Differentialgleichungen, Thesis, Universität Rostock, 1973.

    Google Scholar 

  47. Google Scholar 

  48. G. Wildenhain, Eine Charakterisierung von Mengen der Kapazität Null im R1. Math. Nachr. 32, 269–275 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Wildenhain, Potentialtheorie linearer elliptischer Differentialgleichungen beliebiger Ordnung. Berlin 1968.

    MATH  Google Scholar 

  50. G. Wildenhain, Vergleich verschiedener Kapazitätsbegriffe. Kolloquium “Eliptische Differentialgleichungen I”, Berlin 1970, S. 179–190.

    Google Scholar 

  51. G. Wildenhain, Das “feine” Dirichlet-Problem für elliptische Differentialgleichungen beliebiger Ordnung. Kolloquium “Elliptische Differentialglei- chungen II”, Berlin 1971, S. 247–264.

    Google Scholar 

  52. G. Wildenhain, Über das Randverhalten der Hilbertraumlösung des Dirichlet-Problems. Math. Nachr. 48, 227–235 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  53. Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Lecture Notes in Mathematics, vol. 316, Berlin-Heidelberg-New York 1973.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Sektion Mathematik, GDR, MLU Halle, Germany

    Gottfried Anger

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  1. Gottfried Anger
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Editors and Affiliations

  1. Mathematical Institute of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia

    Josef Král

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© 1975 Academia, Publishing House of the Czechoslovak Academy of Sciences

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Anger, G. (1975). Direct and Inverse Problems in Potential Theory. In: Král, J. (eds) Nonlinear Evolution Equations and Potential Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-4425-4_2

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  • DOI: https://doi.org/10.1007/978-1-4613-4425-4_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-4427-8

  • Online ISBN: 978-1-4613-4425-4

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