no. 3
Geometric conditions which imply compactness of the ∂ ¯-Neumann operator
Straube, Emil1
1 Department of Mathematics, Texas A\&M University, College Station, TX 77843, (USA)
Annales de l'Institut Fourier, Volume 54 (2004) no. 3, pp. 699-710.

For smooth bounded pseudoconvex domains in ℂ 2 , we provide geometric conditions on the boundary which imply compactness of the ∂ ¯-Neumann operator. It is noteworthy that the proof of compactness does not proceed via verifying the known potential theoretic sufficient conditions.

On donne, pour les domaines lisses bornés pseudoconvexes de ℂ 2 , des conditions géométriques concernant le bord qui entraînent la compacité de l’opérateur ∂ ¯-Neumann. Il est remarquable que la preuve de la compacité ne procède pas par verification des conditions suffisantes bien connues de type théorie du potentiel.

DOI: 10.5802/aif.2030
Classification: 32W05
Keywords: $\overline{\partial }$-Neumann operator, compactness, geometric conditions
Mot clés : opérateur $\overline{\partial }$-Neumann, compacité, conditions géométriques
Straube, Emil 1

1 Department of Mathematics, Texas A\&M University, College Station, TX 77843, (USA) 
@article{AIF_2004__54_3_699_0,
     author = {Straube, Emil},
     title = {Geometric conditions which imply compactness of the ${\overline{\partial }}${-Neumann} operator},
     journal = {Annales de l'Institut Fourier},
     pages = {699--710},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {3},
     year = {2004},
     doi = {10.5802/aif.2030},
     zbl = {1061.32028},
     mrnumber = {2097419},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2030/}
}
TY  - JOUR
AU  - Straube, Emil
TI  - Geometric conditions which imply compactness of the ${\overline{\partial }}$-Neumann operator
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 699
EP  - 710
VL  - 54
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2030/
DO  - 10.5802/aif.2030
LA  - en
ID  - AIF_2004__54_3_699_0
ER  - 
%0 Journal Article
%A Straube, Emil
%T Geometric conditions which imply compactness of the ${\overline{\partial }}$-Neumann operator
%J Annales de l'Institut Fourier
%D 2004
%P 699-710
%V 54
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2030/
%R 10.5802/aif.2030
%G en
%F AIF_2004__54_3_699_0
Straube, Emil. Geometric conditions which imply compactness of the ${\overline{\partial }}$-Neumann operator. Annales de l'Institut Fourier, Volume 54 (2004) no. 3, pp. 699-710. doi : 10.5802/aif.2030. https://aif.centre-mersenne.org/articles/10.5802/aif.2030/

[1] Harold P. Boas Small sets of infinite type are benign for the ∂ ¯-Neumann problem, Proceedings of the American Math. Soc, Volume 103 (1988), pp. 569-578 | MR | Zbl

[2] Harold P. Boas; Emil J. Straube; M. Schneider and Y.-T. Siu, eds Global regularity of the ∂ ¯ -Neumann problem: a survey of the L 2 -Sobolev theory, Several Complex Variables (MSRI Publications), Volume 37 (1999) | Zbl

[3] David W. Catlin Boundary behavior of holomorphic functions on weakly pseudoconvex domains (1978) (Princeton University Ph.D. Thesis) | Zbl

[4] David W. Catlin; Y.-T. Siu ed. Global regularity of the ∂ ¯-Neumann problem, Complex Analysis of Several Variables (Proc. Symp. Pure Math.), Volume 41 (1984), pp. 39-49 | Zbl

[5] David W. Catlin Subelliptic estimates for the ∂ ¯-Neumann problem on pseudoconvex domains, Annals of Mathematics (2), Volume 126 (1987), pp. 131-191 | MR | Zbl

[6] So-Chin Chen; Mei-Chi Shaw Partial Differential Equations in Several Complex Variables, Studies in Advanced Mathematics, American Mathematical Society/International Press, 2001 | MR | Zbl

[7] Michael Christ; Siqi Fu Compactness in the ∂ ¯-Neumann problem, magnetic Schrödinger operators, and the Aharonov--Bohm effect (2003) (preprint) | MR | Zbl

[8] John P. D'Angelo Several Complex Variables and the Geometry of Real Hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993 | MR | Zbl

[9] M. Derridj Regularité pour ∂ ¯ dans quelques domaines faiblement pseudo-convexes, Journal of Differential Geometry, Volume 13 (1978), pp. 559-576 | MR | Zbl

[10] G.B. Folland; J.J. Kohn The Neumann Problem for the Cauchy--Riemann Complex, Annals of Mathematics Studies, 75, Princeton University Press, 1972 | MR | Zbl

[11] Siqi Fu; Emil J. Straube Compactness of the ∂ ¯-Neumann problem on convex domains, Journal of Functional Analysis, Volume 159 (1998), pp. 629-641 | MR | Zbl

[12] Siqi Fu; Emil J. Straube; J. McNeal ed. Compactness in the ∂ ¯-Neumann problem, Complex Analysis and Geometry, Volume 9 (2001), pp. 141-160 | Zbl

[13] Siqi Fu; Emil J. Straube Semi-classical analysis of Schrödinger operators and compactness in the ∂ ¯-Neumann problem, Journal of Math. Analysis and Applications, Volume 271 (2002) no. 1, pp. 267-282 | MR | Zbl

[13] Siqi Fu; Emil J. Straube Semi-classical analysis of Schrödinger operators and compactness in the ∂ ¯-Neumann problem. (correction), J. Math. Anal. Appl., Volume 280 (2003) no. 1, pp. 195-196 | MR | Zbl

[14] Torsten Hefer; Ingo Lieb On the compactness of the ∂ ¯-Neumann operator, Ann. Fac. Sci. Toulouse Math (6), Volume 9 (2000), pp. 415-432 | Numdam | MR | Zbl

[15] Peter Matheos A Hartogs domain with no analytic discs in the boundary for which the ∂ ¯-Neumann problem is not compact (1997) (University of California Los Angeles Ph.D. Thesis)

[16] Jeffery D. McNeal A sufficient condition for compactness of the ∂ ¯-Neumann problem, Journal of Functional Analysis, Volume 195 (2002), pp. 190-205 | MR | Zbl

[17] Nessim Sibony Une classe de domaines pseudoconvexes, Duke Math. Journal, Volume 55 (1987), pp. 299-319 | MR | Zbl

[18] Emil J. Straube Plurisubharmonic functions and subellipticity of the ∂ ¯ -Neumann problem on non-smooth domains, Mathematical Research Letters, Volume 4 (1997), pp. 459-467 | MR | Zbl

[19] William P. Ziemer Weakly Differentiable Functions, Graduate Texts in Mathematics, 120, Springer-Verlag, 1989 | MR | Zbl

Cited by Sources: