functional calculus for an elliptic operator on a half-space with general boundary conditions

Giovanni Dore; Alberto Venni

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 3, page 487-543
  • ISSN: 0391-173X

Abstract

top
Let be the realization () of a differential operator on with general boundary conditions (). Here is a homogeneous polynomial of order in complex variables that satisfies a suitable ellipticity condition, and for is a homogeneous polynomial of order ; it is assumed that the usual complementing condition is satisfied. We prove that is a sectorial operator with a bounded functional calculus.

How to cite

top

Dore, Giovanni, and Venni, Alberto. "$H^\infty $ functional calculus for an elliptic operator on a half-space with general boundary conditions." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.3 (2002): 487-543. <http://eudml.org/doc/84479>.

@article{Dore2002,
abstract = {Let $A$ be the $L^p$ realization ($1&lt;p&lt;\infty $) of a differential operator $P(D_x,D_t)$ on $\mathbb \{R\}^n\times \mathbb \{R\}^+$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$$B_k$ is a homogeneous polynomial of order $m_k&lt;2m$; it is assumed that the usual complementing condition is satisfied. We prove that $A$ is a sectorial operator with a bounded $H^\infty $ functional calculus.},
author = {Dore, Giovanni, Venni, Alberto},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {-maximal regularity; Cauchy problem; elliptic differential operator with constant coefficients},
language = {eng},
number = {3},
pages = {487-543},
publisher = {Scuola normale superiore},
title = {$H^\infty $ functional calculus for an elliptic operator on a half-space with general boundary conditions},
url = {http://eudml.org/doc/84479},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Dore, Giovanni
AU - Venni, Alberto
TI - $H^\infty $ functional calculus for an elliptic operator on a half-space with general boundary conditions
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 3
SP - 487
EP - 543
AB - Let $A$ be the $L^p$ realization ($1&lt;p&lt;\infty $) of a differential operator $P(D_x,D_t)$ on $\mathbb {R}^n\times \mathbb {R}^+$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$$B_k$ is a homogeneous polynomial of order $m_k&lt;2m$; it is assumed that the usual complementing condition is satisfied. We prove that $A$ is a sectorial operator with a bounded $H^\infty $ functional calculus.
LA - eng
KW - -maximal regularity; Cauchy problem; elliptic differential operator with constant coefficients
UR - http://eudml.org/doc/84479
ER -

References

top
  1. [1] S. Agmon – A. Douglis – L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. Zbl0093.10401MR125307
  2. [2] M. S. Agranovich – M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, (Russian), Uspehi Mat. Nauk 19 n. 3 (1964), 53-161; translated in: Russian Math. Surveys 19 n. 3 (1964), 53-157. Zbl0137.29602MR192188
  3. [3] H. Amann – M. Hieber – G. Simonett, Bounded -calculus for elliptic operators, Differential Integral Equations 7 (1994), 613-653. Zbl0799.35060MR1270095
  4. [4] W. Arendt – A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory 38 (1997), 87-130. Zbl0879.35041MR1462017
  5. [5] M. Cowling – I. Doust – A. McIntosh – A. Yagi, Banach space operators with a bounded functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51-89. Zbl0853.47010MR1364554
  6. [6] J. Diestel – H. Jarchow – A. Tonge, “Absolutely Summing Operators”, Cambridge Studies in Advanced Mathematics vol. 43, Cambridge University Press, Cambridge, 1995. Zbl0855.47016MR1342297
  7. [7] G. Dore – A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189-201. Zbl0615.47002MR910825
  8. [8] G. Dore – A. Venni, functional calculus for sectorial and bisectorial operators, preprint. Zbl1097.47017MR2110093
  9. [9] N. Dunford – J. T. Schwartz, “Linear Operators. Part I”, Pure and Applied Mathematics vol. 7, Interscience Publishers, New York, 1958. Zbl0084.10402MR117523
  10. [10] X. T. Duong, functional calculus of elliptic operators with coefficients on spaces of smooth domains, J. Austral. Math. Soc. Ser. A 48 (1990), 113-123. Zbl0708.35029MR1026842
  11. [11] X. T. Duong, functional calculus of second order elliptic partial differential operators on spaces, In: “Miniconference on Operators in Analysis (Sydney, 1989)”, I. Doust – B. Jefferies – C. Li – A. McIntosh (eds.), Proc. Centre Math. Anal. A.N.U. vol. 24, A.N.U., Canberra, 1990, pp. 91-102. Zbl0709.47015MR1060114
  12. [12] X. T. Duong – A. McIntosh, Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients, J. Geom. Anal. 6 (1996), 181-205. Zbl0897.47041MR1469121
  13. [13] X. T. Duong – E. M. Ouahabaz, Complex multiplicative perturbations of elliptic operators: heat kernel bounds and holomorphic functional calculus, Differential Integral Equations 12 (1999), 395-418. Zbl1008.47020MR1674426
  14. [14] X. T. Duong – G. Simonett, -calculus for elliptic operators with nonsmooth coefficients, Differential Integral Equations 10 (1997), 201-217. Zbl0892.47017MR1424807
  15. [15] E. Franks – A. McIntosh, Discrete quadratic estimates and holomorphic functional calculi in Banach spaces, Bull. Austral. Math. Soc. 58 (1998), 271-290. Zbl0942.47011MR1642055
  16. [16] Y. Giga – H. Sohr, Abstract -estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal. 102 (1991), 72-94. Zbl0739.35067MR1138838
  17. [17] G. H. Hardy – J. E. Littlewood – G. Pólya, “Inequalities”, Cambridge University Press, Cambridge, 1934. Zbl0010.10703JFM60.0169.01
  18. [18] M. Hieber – J. Prüss, Functional calculi for linear operators in vector-valued -spaces via the transference principle, Adv. Differential Equations 3 (1998), 847-872. Zbl0956.47008MR1659281
  19. [19] N. J. Kalton – L. Weis, The -calculus and sums of closed operators, Math. Ann. 321 (2001) 319-345. Zbl0992.47005MR1866491
  20. [20] F. Lancien – G. Lancien – C. Le Merdy, A joint functional calculus for sectorial operators with commuting resolvents, Proc. London Math. Soc. (3) 77 (1998), 387-414. Zbl0904.47015MR1635157
  21. [21] A. McIntosh – A. Nahmod, Heat kernel estimates and functional calculi of , Math. Scand. 87 (2000), 287-319. Zbl1069.35023MR1795749
  22. [22] J. Prüss – H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. 203 (1990), 429-452. Zbl0665.47015MR1038710
  23. [23] J. Prüss – H. Sohr, Imaginary powers of elliptic second order differential operators in -spaces, Hiroshima Math. J. 23 (1991), 161-192. Zbl0790.35023MR1211773
  24. [24] R. T. Seeley, Complex powers of an elliptic operator, In: “Singular Integrals (Chicago, 1966)”, Proc. Simpos. Pure Math. vol. 10, American Mathematical Society, Providence, 1967, pp. 288-307. Zbl0159.15504MR237943
  25. [25] R. T. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math. 91 (1969), 889-920. Zbl0191.11801MR265764
  26. [26] R. T. Seeley, Norms and domains of the complex powers, Amer. J. Math. 93 (1971), 299-309. Zbl0218.35034MR287376
  27. [27] H. Sohr – G. Thäter, Imaginary powers of second order differential operators and -Helmholtz decomposition in the infinite cylinder, Math. Ann. 311 (1998), 577-602. Zbl0911.35088MR1637935
  28. [28] V. A. Solonnikov, On general boundary problems for systems which are elliptic in the sense of A. Douglis and L. Nirenberg. I, (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 665-706; translated in: Amer. Math. Soc. Transl. Ser. 2 56 (1966), 193-232. Zbl0175.11703MR211070
  29. [29] Ž. Štrkalj – L. Weis, On operator-valued Fourier multiplier theorems, preprint. Zbl1209.42005
  30. [30] H. Triebel, “Interpolation Theory, Function Spaces, Differential Operators”, North-Holland Mathematical Library vol. 18, North-Holland Publishing Co., Amsterdam, 1978. Zbl0387.46032MR503903
  31. [31] A. Venni, Marcinkiewicz and Mihlin multiplier theorems, and R-boundedness, preprint. Zbl1031.43002MR2013204

NotesEmbed ?

top

You must be logged in to post comments.