Abstract
We give complete algebraic characterizations of the Lp-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form \(\partial_{h}({\mathop{\cal A}\nolimits}^{hk}(x)\partial_{k})\), where \({\mathop{\cal A}\nolimits}^{hk}(x)\) are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is Lp-dissipative if and only if
ν being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the Lp-dissipativity of the operator \(\partial_{h} ({\mathop{\cal A}\nolimits}^{h}(x)\partial_{h})\), where \({\mathop{\cal A}\nolimits}^{h}(x)\) are m× m matrices with complex L1loc entries, and we describe the maximum angle of Lp-dissipativity for this operator.
Keywords: Lp-dissipativity, Algebraic conditions, Elasticity system
Mathematics Subject Classification (2000): 47D03, 47D06, 47B44, 74B05
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
1. Amann, H.: Dual semigroups and second order elliptic boundary value problems. Israel J. Math. 45, 225–254 (1983)
2. Auscher, P., Barthélemy, L., Bénilan, P., Ouhabaz, El M.: Absence de la Linfty-contractivité pour les semi-groupes associés auz opérateurs elliptiques complexes sous forme divergence. Poten. Anal. 12, 169–189 (2000)
3. Brezis, H., Strauss, W.A.: Semi-linear second order elliptic equations in L1. J. Math. Soc. Japan 25, 565–590 (1973)
4. Cialdea, A., Maz'ya, V.G.: Criterion for the Lp-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl. 84, 1067–1100 (2005)
5. Daners, D.: Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217, 13–41 (2000)
6. Davies, E.B.: One-parameter semigroups. Academic Press, London-New York (1980)
7. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge, U.K. (1989)
8. Davies, E.B.: Lp spectral independence and L1 analiticity. J. London Math. Soc. (2) 52, 177–184 (1995)
9. Davies, E.B.: Uniformly elliptic operators with measurable coefficients. J. Funct. Anal. 132, 141–169 (1995)
10. Fattorini, H.O.: The Cauchy Problem. Encyclopedia of Mathematics and its Applications 18, Addison-Wesley Publishing Co., Reading, Mass. (1983)
11. Fattorini, H.O.: On the angle of dissipativity of ordinary and partial differential operators. In Zapata, G.I. (ed.), Functional Analysis, Holomorphy and Approximation Theory, II, Math. Studies 86, North-Holland, Amsterdam, 85–111 (1984)
12. Fichera, G.: Linear elliptic differential systems and eigenvalue problems. Lecture Notes in Math., 8, Springer-Verlag, Berlin Heidelberg (1965)
13. Karrmann, S.: Gaussian estimates for second order operators with unbounded coefficients. J. Math. Anal. Appl. 258,320–348 (2001)
14. Kovalenko, V., Semenov, Y.: C0-semigroups in Lp(mathbbRd) and hat C (mathbb Rd) spaces generated by the differential expression Δ + b ⋅ nabla. Theory Probab. Appl. 35, 443–453 (1990)
15. Kresin, G.I., Maz'ya, V.G.: Criteria for validity of the maximum modulus principle for solutions of linear parabolic systems. Ark. Mat. 32, 121–155 (1994)
16. Langer, M.: Lp-contractivity of semigroups generated by parabolic matrix differential operators. In: The Maz'ya Anniversary Collection, Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications, Birkhäuser, Boston 307–330 (1999)
17. Langer, M., Maz'ya, V.: On Lp-contractivity of semigroups generated by linear partial differential operators. J. Funct. Anal. 164, 73–109 (1999)
18. Liskevich, V.: On C0-semigroups generated by elliptic second order differential expressions on Lp-spaces. Differ. Integral Equ. 9, 811–826 (1996)
19. Liskevich, V.A., Semenov, Yu.A.: Some problems on Markov semigroups. In: Demuth, M. (ed.) et al., Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras. Berlin: Akademie Verlag. Math. Top. 11, 163–217 (1996)
20. Liskevich, V., Sobol, Z., Vogt, H.: On the L p -theory of C0 semigroups associated with second order elliptic operators. II. J. Funct. Anal. 193, 55–76 (2002)
21. Maz'ya, V., Sobolevskii, P.: On the generating operators of semigroups (Russian). Uspekhi Mat. Nauk 17, 151–154 (1962)
22. Metafune, G., Pallara, D., Prüss, J., Schnaubelt, R.: Lp-theory for elliptic operators on mathbb Rd with singular coefficients. Z. Anal. Anwendungen 24, 497–521 (2005)
23. Okazawa, N.: Sectorialness of second order elliptic operators in divergence form. Proc. Amer. Math. Soc. 113, 701–706 (1991)
24. Ouhabaz, E.M.: Gaussian upper bounds for heat kernels of second-order elliptic operators with complex coefficients on arbitrary domains. J. Operator Theory 51, 335–360 (2004)
25. Ouhabaz, E.M.: Analysis of heat equations on domains. London Math. Soc. Monogr. Ser. 31, Princeton Univ. Press, Princeton (2005)
26. Robinson, D.W.: Elliptic operators on Lie groups. Oxford University Press, Oxford (1991)
27. Sobol, Z., Vogt, H.: On the L p -theory of C0 semigroups associated with second order elliptic operators. I. J. Funct. Anal. 193, 24–54 (2002)
28. Strichartz, R.S.: Lp contractive projections and the heat semigroup for differential forms. J. Funct. Anal. 65, 348–357 (1986)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cialdea, A., Maz’ya, V. Criteria for the Lp-dissipativity of systems of second order differential equations. Ricerche mat. 55, 73–105 (2006). https://doi.org/10.1007/s11587-006-0014-x
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11587-006-0014-x