no. 3
Geometric conditions which imply compactness of the ∂ ¯-Neumann operator
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: ∂ ¯-Neumann operator, compactness, geometric conditions 
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Straube, Emil. Geometric conditions which imply compactness of the $$-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/

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

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

[9] M. Derridj Regularité pour $\bar\partial$ dans quelques domaines faiblement pseudo-convexes, Journal of Differential Geometry, Tome 13 (1978), pp. 559-576 | Zbl

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

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

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

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

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

[15] Peter Matheos A Hartogs domain with no analytic discs in the boundary for which the $\bar\partial$-Neumann problem is not compact (1997) (University of California Los Angeles Ph.D. Thesis) | MR | Zbl

[18] Emil J. Straube Plurisubharmonic functions and subellipticity of the $\hbox\bar\partial$-Neumann problem on non-smooth domains, Mathematical Research Letters, Tome 4 (1997), pp. 459-467 | MR | Zbl

[11] Siqi Fu; Emil J. Straube Compactness of the $\bar\partial$-Neumann problem on convex domains, Journal of Functional Analysis, Tome 159 (1998), pp. 629-641 | Zbl

[2] Harold P. Boas; Emil J. Straube; M. Schneider and Y.-T. Siu, eds Global regularity of the $\bar\partial$-Neumann problem: a survey of the $\cl L^¨2$-Sobolev theory, Several Complex Variables (MSRI Publications) Tome 37 (1999) | MR | Zbl

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

[12] Siqi Fu; Emil J. Straube; J. McNeal ed. Compactness in the $\bar\partial$-Neumann problem, Complex Analysis and Geometry, Tome 9 (2001), pp. 141-160

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

[16] Jeffery D. McNeal A sufficient condition for compactness of the $\bar\partial$-Neumann problem, Journal of Functional Analysis, Tome 195 (2002), pp. 190-205 | MR | Zbl

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

Siqi Fu; Emil J. Straube Semi-classical analysis of Schrödinger operators and compactness in the $\bar\partial$-Neumann problem. (correction), J. Math. Anal. Appl., Tome 280 (2003) no. 1, p. 195-196 | MR | Zbl

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