Baillon’s theorem on maximal regularity. (English) Zbl 0802.47039
Summary: The aim of this note is to give a proof of Baillon’s theorem on maximal regularity, though it is in some sense a negative result (it states that for abstract Cauchy problems maximal regularity can occur only in very special cases), it is commonly accepted that it is important. Many people believe that its proof is very complicated. This might be due to the fact that J. B. Baillon’s note in C. R. Acad. Sci., Paris, Sér. A 290, 757-760 (1980; Zbl 0436.47027) is rather short and sometimes difficult to understand. The proof outlined here follows basically Baillon’s lines. However it is simplified and (hopefully) easier to understand.
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
Citations:
Zbl 0436.47027References:
| [1] | J.B. Baillon, Caractère borné de certains générateurs de semigroupes linéaires dans les espaces de Banach, C.R. Acad. Sc. Paris 290 (1980), pp. 757-760. · Zbl 0436.47027 |
| [2] | C. Bessaga, A. Pelczy?ski, On bases and unconditional convergence of series in Banach spaces, Studia Math. XVII (1958), pp. 329-396. |
| [3] | Ph. Clément (et al.), One-Parameter Semigroups, CWI Monographs 5, North-Holland, Amsterdam, 1987. |
| [4] | N. Dunford, J.T. Schwartz, Linear Operators, I, Wiley, New York, 1958. |
| [5] | E. Hille, On the differentiability of semigroups of operators, Acta Sc. Math. (Szeged) 12 (1950), pp. 19-24. · Zbl 0035.35802 |
| [6] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. · Zbl 0516.47023 |
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