Sectorialness of second order elliptic operators in divergence form

Okazawa, Noboru

doi:10.1090/S0002-9939-1991-1072347-4

Sectorialness of second order elliptic operators in divergence form
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Proc. Amer. Math. Soc. 113 (1991), 701-706 Request permission

Abstract:

A sectorial estimate is given to second order linear elliptic differential operators of divergence form. The estimate is a slight improvement of Pazy’s. The obtained constant depends on $p$ of the space ${L^p}(\Omega )(1 < p < \infty )$ and does not depend on the operators themselves. The same constant has appeared in the sectorial estimate for second order linear ordinary differential operators due to Fattorini. The result is in connection with Stein’s estimate of the analytic semigroups generated by linear elliptic differential operators.

References

Hector O. Fattorini, The Cauchy problem, Encyclopedia of Mathematics and its Applications, vol. 18, Addison-Wesley Publishing Co., Reading, Mass., 1983. With a foreword by Felix E. Browder. MR 692768
Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Rainer Hempel and Jürgen Voigt, On the $L_p$-spectrum of Schrödinger operators, J. Math. Anal. Appl. 121 (1987), no. 1, 138–159. MR 869525, DOI 10.1016/0022-247X(87)90244-7
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Dan Pascali and Silviu Sburlan, Nonlinear mappings of monotone type, Martinus Nijhoff Publishers, The Hague; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978. MR 531036
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
H. O. Fattorini, On the angle of dissipativity of ordinary and partial differential operators, Functional analysis, holomorphy and approximation theory, II (Rio de Janeiro, 1981) North-Holland Math. Stud., vol. 86, North-Holland, Amsterdam, 1984, pp. 85–111. MR 771324, DOI 10.1016/S0304-0208(08)70824-7
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244