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Parabolic pseudo-differential boundary problems and applications

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Microlocal Analysis and Applications

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References

  1. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623–727.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. S. Agranovič and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of a general type, Usp. Mat. Nauk 19 (1963), 53–161; English transl., Russ. Math. Surveys 19 (1963), 53–159.

    Google Scholar 

  3. J. Bergh and J. Löfström, “Interpolation spaces,” Springer Verlag, Berlin, New York, 1976.

    Book  MATH  Google Scholar 

  4. L. Boutet de Monvel, Comportement d'un opérateur pseudo-différentiel sur une variété à bord, I–II, J. d'Analyse Math. 17 (1966), 241–304.

    Article  MATH  MathSciNet  Google Scholar 

  5. —, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Eskin, “Boundary value problems for elliptic pseudodifferential equations,” AMS, Providence, R. I., 1981.

    MATH  Google Scholar 

  7. L. Frank and W. D. Wendt, Coercive singular perturbations III: Wiener-Hopf operators, J. d'Analyse Math. 43 (1984), 88–135.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Franke, Besov-Triebel-Lizorkin spaces and boundary value problems, “Seminar Analysis,” Karl-Weierstrass-Institut für Math., Berlin DDR, 1984/85, pp. 89–104.

    Google Scholar 

  9. J. Franke, “Elliptische Randwertprobleme in Besov-Triebel-Lizorkin-Raümen,” Dissertation, Friedrich-Schiller-Universität Jena 1986.

    Google Scholar 

  10. A. Friedman, “Partial differential equations,” Holt, Rinehart and Winston, New York, 1969.

    MATH  Google Scholar 

  11. P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rat. Mech. Anal. 25 (1967), 40–63.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Grubb, Singular Green operators and their spectral asymptotics, Duke Math. J. 51 (1984), 477–528.

    Article  MATH  MathSciNet  Google Scholar 

  13. —, “Functional calculus of pseudo-differential boundary problems,” Progress in Math. Vol. 65, Birkhäuser, Boston, 1986.

    Book  MATH  Google Scholar 

  14. —, Pseudo-differential boundary problems in L p spaces, Comm. Part. Diff. Eq. 15 (1990), 289–340.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Grubb and L. Hörmander, The transmission property, Math. Scand. 67 (1990), 273–289.

    MATH  MathSciNet  Google Scholar 

  16. G. Grubb and V. A. Solonnikov, Reduction of basic initial-boundary value problems for the Stokes equation to initial-boundary value problems for systems of pseudodifferential equations, Zap. Nauchn. Sem. L.O.M.I. 163 (1987), 37–48; = J. Soviet Math. 49 (1990), 1140–1147.

    MATH  Google Scholar 

  17. A pseudo-differential treatment of general inhomogeneous initial-boundary value problems for the Navier-Stokes equation, Exposé no. 3, “Proc. of Journées Equations Dér. Part., St. Jean de Monts 1988,” Ecole Polytechnique, Palaiseau, 1988, 8 pp.

    Google Scholar 

  18. —, Solution of parabolic pseudo-differential initial-boundary value problems, J. Diff. Eq. 87 (1990), 256–304.

    Article  MATH  MathSciNet  Google Scholar 

  19. —, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods, to appear in Math. Scand. (available as Copenh. Math. Dept Preprint Series no. 14, 1990).

    Google Scholar 

  20. D. Huet, “Décomposition spectrale et opérateurs,” Presses Universitaires de France, Paris, 1977.

    Google Scholar 

  21. L. Hörmander, “Linear partial differential operators,” Springer Verlag, Berlin, New York, 1963.

    Book  MATH  Google Scholar 

  22. —, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. 83 (1966), 129–209.

    Article  MATH  MathSciNet  Google Scholar 

  23. —, “The analysis of linear partial differential operators, vol. I–IV,” Springer Verlag, Berlin, New York, 1983–85.

    Book  Google Scholar 

  24. T. Kato, “Perturbation theory for linear operators,” Springer Verlag, Berlin, New York, 1966.

    MATH  Google Scholar 

  25. I. Lasiecka and R. Triggiani, Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations, SIAM J. Control Optim. 21 (1983), 766–803.

    Article  MATH  MathSciNet  Google Scholar 

  26. O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltzeva, “Linear and Quasilinear Equations of Parabolic Type,” AMS, Providence, 1968.

    Google Scholar 

  27. J. L. Lions and E. Magenes, “Problèmes aux limites non homogènes et applications, vol. 1–2,” Editions Dunod, Paris, 1968.

    MATH  Google Scholar 

  28. L. Nirenberg, “Lectures on linear partial differential equations,” AMS Regional Conference Proceedings no. 17, Providence, 1973.

    Google Scholar 

  29. M. Pedersen, Pseudo-differential perturbations and stabilization of distributed parameter systems: Dirichlet feedback conirol problems, SIAM J. Contr. Opt. 29 (1991), 222–252.

    Article  MATH  Google Scholar 

  30. V. T. Purmonen, Parabolic pseudo-differential initial-boundary value problems, Math. Scand. 65 (1989), 221–244.

    MATH  MathSciNet  Google Scholar 

  31. S. Rempel and B.-W. Schulze, “Index theory of elliptic boundary problems,” Akademie Verlag, Berlin, 1982.

    MATH  Google Scholar 

  32. —, Parametrices and boundary symbolic calculus for elliptic boundary problems without the transmission property, Math. Nachr. 105 (1982), 45–149.

    Article  MATH  MathSciNet  Google Scholar 

  33. —, Complez powers for pseudo-differential boundary problems II, Math. Nachr. 116 (1984), 269–314.

    Article  MATH  MathSciNet  Google Scholar 

  34. Sh. Sakhaev and V. A. Solonnikov, Estimates of solutions of a boundary value problem in magnetohydrodynamics, Proc. Steklov Math. Inst. 127 (1965), 76–92.

    Google Scholar 

  35. R. T. Seeley, Extension of C functions defined in a half space, Proc. AMS 15 (1964), 625–626.

    MATH  MathSciNet  Google Scholar 

  36. —, Singular integrals and boundary problems, Amer. J. Math. 88 (1966), 781–809.

    Article  MATH  MathSciNet  Google Scholar 

  37. —, The resolvent of an elliptic boundary problem, Amer. J. Math. 91 (1969), 889–920.

    Article  MATH  MathSciNet  Google Scholar 

  38. F. Treves, “Introduction to pseudodifferential and Fourier integral operators, vol. 1–2,” Plenum Press, New York, 1980.

    Google Scholar 

  39. H. Triebel, “Interpolation theory, function spaces, differential operators,” North-Holland Publ. Co., Amsterdam, New York, 1978.

    Google Scholar 

  40. M. I. Vishik and G. I. Eskin, Elliptic equations in convolution in a bounded domain and their applications, Uspehi Mat. Nauk 22 (1957), 15–76; = Russian Math. Surveys 22 (1967), 13–75.

    Google Scholar 

  41. M. I. Vishik and L. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameters, Uspekhi Mat. Nauk 12 (1957), 3–122; = AMS Transl. (2) 20 (1962), 239–264.

    MATH  MathSciNet  Google Scholar 

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Grubb, G. (1991). Parabolic pseudo-differential boundary problems and applications. In: Cattabriga, L., Rodino, L. (eds) Microlocal Analysis and Applications. Lecture Notes in Mathematics, vol 1495. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085122

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  • DOI: https://doi.org/10.1007/BFb0085122

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