Abstract
This paper is devoted to the study of contraction semigroups generated by second order linear partial differential operators and is written in collaboration with V. Maz’ya. The paper begins with a brief survey of the results obtained earlier by the author and V. Maz’ya. A class of weakly coupled systems is studied and a close relationship between the generation of (L P)N-contractive semigroups of the corresponding operators and (L 2)N-contractivity of the semigroups generated by some associated operators is obtained. More precisely it is seen that (L 2)N-dissipativity of an associated operator implies (L p)N-dissipativity of the original operator, whereas the converse holds for some subclass of operators, which includes the scalar operators.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Amann, H., Dual Semigroups and Second Order Linear Elliptic Boundary Value Problems, Isr. J. Math. 45 (1983), 225–254.
Brezis, H. and W. A. Strauss, Semi-linear Second-order Elliptic Equations in L 1, J. Math. Soc. Japan 25 (1973), 565–590.
Cycon, H. L., Froese, R. G., Kirsch, W. and B. Simon, Schrödinger Operators, Springer-Verlag Berlin Heidelberg, 1987.
Grisvard, P., Elliptic Problems in Nonsmooth Domains, Pitman Publishing Ltd., London, 1985.
Goldstein, J. A., Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1985.
Hörmander, L., The Analysis of Linear Partial Differential Operators III, Springer-Verlag, Berlin Heidelberg, 1985.
Ladyžhenskaya, O. A., The Boundary Value Problems of Mathematical Physics, Springer-Verlag New York, 1985.
Ladyžhenskaya, O. A., Solonnikov, V. A. and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Providence, R. I., 1968.
Langer, M. and V. G. Maz’ya, On L Pcontractivity of Semigroups Generated by Linear Partial Differential Operators, J. Func. Analysis, To appear.
Kresin, G. I. and V. G. Maz’ya, Criteria for Validity of the Maximum Modulus Principle for Solutions of Linear Parabolic Systems, Ark. Mat. 32 (1994), 121–155.
Maz’ya, V. G. and P. E. Sobolevskiĭ, On Generating Operators of Semigroups (Russian), Uspekhi Mat. Nauk 17:6 (1962), 151–154.
Simon, B., Schrödinger Semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Langer, M. (1999). L p-contractivity of semigroups generated by parabolic matrix differential operators. In: Rossmann, J., Takáč, P., Wildenhain, G. (eds) The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol 109. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8675-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8675-8_20
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9726-6
Online ISBN: 978-3-0348-8675-8
eBook Packages: Springer Book Archive