×

Applications of semigroups of operators to non-elliptic differential operators. (English) Zbl 1064.47504

Summary: This paper treats systematically the semigroup method of non-elliptic differential operators, which was developed in the last ten years. In particular, a review of the applications of regularized semigroups to non-elliptic differential operators with constant coefficients or time-dependent coefficients, parabolic systems, correct systems, abstract differential operators and pseudodifferential operators is given here. It is also shown that the regularized semigroup is an appropriate tool for non-elliptic differential operators and is far superior to the integrated semigroup approach.

MSC:

47D60 \(C\)-semigroups, regularized semigroups
47D62 Integrated semigroups
47F05 General theory of partial differential operators
35P99 Spectral theory and eigenvalue problems for partial differential equations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Hörmander, L., Estimates for translation invariant operators inL p spaces, Acta Math., 1960, 104: 93. · Zbl 0093.11402
[2] de Laubenfels, R., Existence Families, Functional Calculi and Evolution Equations, Berlin: Springer-Verlag, 1994. · Zbl 0815.47042
[3] Zheng, Q., Integrated semigroups and abstract Cauchy problems (in Chinese), Adv. Math, 1992, 21: 257. · Zbl 0759.34045
[4] Hieber, M., Integrated semigroups and differential operators on Lp spaces, Math. Ann., 1991, 291: 1. · Zbl 0724.34067
[5] Kellermann, H., Hieber, M., Integrated semigroups, J. Funct. Anal., 1989, 84: 160. · Zbl 0689.47014
[6] Da Prato, G., Semigruppi regolarizzabili, Ricerche Mat, 1966, 15: 223.
[7] Davies, E. B., Pang, M. M. H., The Cauchy problem and a generalization of the Hille-Yosida theorem, Proc. London Math. Soc, 1987, 55: 181. · Zbl 0651.47026
[8] Mel’nikova, I. V., Filinkov, A. I., Integrated semigroups and C-semigroups, well-posedness and regularization of differential-operator problems, Russian Math. Surveys, 1994, 49(6): 115. · Zbl 0889.47024
[9] Tanaka, N., Okazawa, N., Local C-semigroups and local integrated semigroups, Proc. London Math. Soc., 1990, 61: 63. · Zbl 0728.47030
[10] de Laubenfels, R., C-existcncc families and improperly posed problems, Semesterbericht Funktionalanalysis, Tubingen: 1990, 17: 155.
[11] Zheng, Q., Liu, L., Almost periodic regularized group, semigroups, and cosine functions, J. Math. Anal. Appl., 1996, 197: 90. · Zbl 0888.47020
[12] Zheng, Q., Controllability of a class of linear systems in Banach space, in Proc. Amer. Math. Soc., 1995, 123: 1241. · Zbl 0833.93029
[13] Shaw, S.-Y., Li, Y.-C., On N-times integrated C-cosine functions, in Evolution Equations (eds. Ferreyra, G., Goldstein, G. R., Neubrander, F.), New York: Marcel Dekker, 1994., 393. · Zbl 0935.47032
[14] Kuo, C.-C., Shaw, S.-Y., C-cosine functions and the abstract Cauchy problem I, II, J. Math. Anal. Appl., 1997, 210: 632. · Zbl 0881.34071
[15] Zheng, Q., Lei, Y., Exponentially bounded C-cosine functions of operators, J. Sys. Sci. Math. Scis. (in Chinese), 1996, 16: 242. · Zbl 0897.47039
[16] Arendt, W., Vector-valued Laplace transforms and Cauchy problems, Israel J. Math., 1987, 59: 327. · Zbl 0637.44001
[17] Hieber, M., Laplace transforms and {\(\alpha\)}-times integrated semigroups, Forum Math., 1991, 3: 595. · Zbl 0766.47013
[18] Dunford, N., Schwartz, J. T., Linear Operators I: General Theory. New York: Interscience, 1958. · Zbl 0084.10402
[19] Zheng, Q., Integrated cosine functions, Internat. J. Math. Math. Sci., 1996, 19: 575. · Zbl 0854.47028
[20] Zhang, J. Zheng, Q., On {\(\alpha\)}-times integrated cosine functions, Math. Japon., 1999, 50: 401 · Zbl 0951.47043
[21] Arendt, W., Kellermann, H., Integrated solutions of Volterra integro-differential equations and applications, in: Volterra Integra-differential Equations in Banach Spaces and Applications, (eds. Da Prato, G., Iannelli, M.). Harlow: Longman, 1989, 21. · Zbl 0675.45017
[22] Arendt, W., Sobolev imbedding and integrated semigroups, in Semigroup Theory and Evolution Equations, (eds. Clément, Ph., Mitidieri, E., de Pagter, B.), New York: Marcel Dekker, 1991, 29–40. · Zbl 0762.47013
[23] Hieber, M., Räbiger, F., A remark on the abstract Cauchy problem on spaces of Hölder continuous functions, Proc. Amer. Math. Soc., 1992, 115: 431. · Zbl 0765.34042
[24] Mijatovic, M., Pilipovic, S., {\(\alpha\)}-times integrated semigroups ({\(\alpha\)}R+), J. Math. Anal. Appl., 1997, 210: 790. · Zbl 0882.47017
[25] Hieber, M., Spectral theory and Cauchy problems on Lp-spaces, Math. Z., 1994, 216: 613. · Zbl 0905.47035
[26] Hieber, M., Holderrieth, A., Neubrander, F., Regularized semigroups and systems of linear partial differential equations, Ann. Scuola Norm. Sup. Pisa. 1992, 19: 363. · Zbl 0789.35075
[27] de Laubenfels, R., Matrices of operators and regularized semigroups, Math. Z., 1993, 212: 619. · Zbl 0792.47041
[28] Lei, Y., Yi, W., Zheng, Q., Semigroups of operators and polynomials of generators of bounded strongly continuous groups, Proc. London Math. Soc., 1994, 69: 144. · Zbl 0815.47049
[29] Lei, Y., Zheng, Q., The application of C-semigroups to differential operators in Lp(lon), J. Math. Anal. Appl., 1994, 188: 809. · Zbl 0938.47035
[30] Hieber, M., On linear hyperbolic systems with multiple characteristics, Differential Integral Equations, 1995, 8: 877. · Zbl 0822.47043
[31] Zheng, Q.. Abstract differential operators and Cauchy problems, Tübinger Berichte Funktionalanalysis, 1995, 4: 273.
[32] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, New Jersey: Princeton Univ Press, 1970. · Zbl 0207.13501
[33] Sjöstrand, S., On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa, 1970, 24: 331. · Zbl 0201.14901
[34] Miyachi, A.. On some singular Fourier multipliers, J. Fac Sci. Univ. Tokyo, 1981, 28: 267. · Zbl 0469.42003
[35] Iha, F. T., Schubert, C. F., The spectrum of partial differential operators onL p (\(\mathbb{R}\) n Trans. Amer. Math. Soc., 1970, 152: 215. · Zbl 0219.47044
[36] Tanaka, N., Linear evolution equations in Banach spaces, in Proc London math. Soc., 1991, 63: 657. · Zbl 0746.34036
[37] de Laubentels, R., Simultaneous well-posedness, in Evolution Equations, Control Theory, and Biomathematics (eds. Clé-ment, Ph Lumer, G.), New York: Marcel Dekker, 1993, 101.
[38] Straub, B. On fractional powers of closed operators with polynomially bounded resolvents and applications to the abstract Cauchy problem, Ph D Thesis, Universität Tubingen, 1994. · Zbl 0840.47013
[39] Brenner, P., The Cauchy problem for symmetric hyperbolic systems inL p , Math. Scand., 1966, 19: 27. · Zbl 0154.11304
[40] Brenner, P., The Cauchy problem for systems inL p andL p,u , Ark. Mat., 1973, 11: 75. · Zbl 0256.35006
[41] Hieber, M., Integrated semigroups and the Cauchy problem for systems in Lp-spaces, J. Math. Anal. Appl., 1991, 162: 300. · Zbl 0766.47014
[42] Zheng, Q., Li, Y., Abstract parabolic systems and regularized semigroups, Pacific J. Math. 1998, 182: 183. · Zbl 0915.47036
[43] Zhang, J., Zheng, Q.. Wellposedness of parabolic systems onC {\(\alpha\)} in Nonlinear Partial Differential Equations and Applications, (eds. Guo, B., Yang, D.), Singapore: World Scientific, 1998. 231–234. · Zbl 0982.35046
[44] Hieber, M., An operator-theoretical approach to Dirac’s equation on Lp-spaces, in Semigroup Theory and Evolution Equations (eds. Chément, Ph., de Bagter, B., Mitidieri, E.), New York: Marcel Dekker, 1991, 259–265. · Zbl 0748.47036
[45] Holderrieth, A., Matrix multiplication operators generating one parameter semigroups, Semigroup Forum, 1991, 42: 155. · Zbl 0744.47033
[46] Gelfand, I. M., Shilov, G. E., Generalized Functions III, New York: Academic Press, 1967.
[47] Littman, W., The wave operator and Lp-norms, J. Math. Mech., 1963, 12: 55. · Zbl 0127.31705
[48] Zhang, J., Regularized cosine functions and polynomials of group generators, Sci. Iranica, 1997, 4: 12.
[49] Zhang, J., Zheng, Q., Pseudodifferential operators and regularized cosine functions (in Chinese), Acta Math. Sinica, 1998, 41: 767. · Zbl 1022.47504
[50] de Laubentels, R., Polynomials of generators of integrated semigroups, Proc. Amer. Math. Soc., 1989, 107: 197.
[51] de Laubentels, R., Entire solutions of the abstract Cauchy problem, Semigroup Forum, 1991, 42: 83. · Zbl 0746.47018
[52] Zheng, Q., Lei, Y., Polynomials of the generator of a bounded strongly continuous group (in Chinese), J. Huazhong Univ. Sci. Tech., 1993, 21(5): 171. · Zbl 0788.47037
[53] Lei, Y., Yi, W., Zheng, Q., C-evolution operators and wellposedness of some linear evolution equations, in Applied Functional Analysis. Vol. 1 (ed. Yang, M.), Beijing: International Academic Publishers, 1993, 115–119.
[54] Zheng, Q., Semigroups and weakly elliptic operators, Acta Anal. Funct. Appl., 1999, 1: 40.
[55] Arveson, W. B., On groups of automorphisms of operator algebras, J. Funct. Anal., 1974, 15: 217. · Zbl 0296.46064
[56] Davies, E. B., One-parameter Semigroups, London: Academic Press, 1980. · Zbl 0457.47030
[57] Hieber, M.,L p spectra of pseudodifferential operators generating integrated semigroups, Trans. Amer. Math. Soc., 1995, 347: 4023. · Zbl 0847.47027
[58] Zhang, J., Regularized operator families and applications to differential operators, Ph D Thesis, Huazhong University of Science and Technology, 1996.
[59] deLaubenfels, R., Lei, Y., Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators, Abstract Appl. Anal., 1997. 2: 121. · Zbl 0937.47015
[60] Zhang, J.. Zheng, Q., Pseudodifferential operators and regularized semigroups, Chinese J. Contemporary Math., 1998, 19: 387. · Zbl 0928.47030
[61] Zhang, J., Regularized semigroups generated by pseudodifferential operators onC {\(\alpha\)} Ann. Diff. Eqs., 1998, 14: 427.
[62] El-Mennaoui, O., Traces des semi-groupes holomorphes singuliers à l’origine et comportement asymptotique, Ph D Thesis, Université de Franche-Comté, 1992.
[63] Zheng, Q., Zhang, J., Gaussian estimates and regularized groups, Proc. Amer. Math. Soc., 1999, 127: 1089. · Zbl 0912.47019
[64] Pang, M. M. H., Resolvent estimates Schrödinger operators in Lp/(\(\mathbb{R}\) n ) and the theory of exponentially bounded C-semigroups, Semigroup Forum, 1990, 41: 97. · Zbl 0739.47017
[65] Boyadzhiev, K., de Laubenfels, R., Boundary values of holomorphic semigroups, Proc. Amer. Math. Soc., 1993, 118: 113. · Zbl 0797.47023
[66] Balabane, M., Emamirad, H.,L p estimates for Schrödinger evolution equations, Trans. Amer. Math. Soc., 1985, 118: 357. · Zbl 0588.35029
[67] Keyantuo, V., A note on interpolation of semigroups, Proc. Amer. Math. Soc., 1995, 123: 2123. · Zbl 0832.47032
[68] El-Mennaoui, O., Keyantuo, V., On the Schrödinger equation inL p spaces, Math. Ann., 1996, 304: 293. · Zbl 0842.35024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.