Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The $\overline{\partial }$ problem on domains
with piecewise smooth boundaries
with applications

Authors: Joachim Michel and Mei-Chi Shaw
Journal: Trans. Amer. Math. Soc. 351 (1999), 4365-4380
MSC (1991): Primary 35N05, 35N10, 32F10
Published electronically: July 9, 1999
MathSciNet review: 1675218
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega$ be a bounded domain in $\mathbb C^n$ such that $\Omega$ has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation

\begin{equation*}\overline{\partial}u=\alpha\quad \text{in}\quad \Omega\tag{0.1} \end{equation*}

where $\alpha$ is a smooth $\overline{\partial}$-closed $(p,q)$ form with coefficients $C^\infty$ up to the bundary of $\Omega$, $0\le p\le n$ and $1\le q\le n$. In particular, Equation (0.1) is solvable with $u$ smooth up to the boundary (for appropriate degree $q)$ if $\Omega$ satisfies one of the following conditions:

$\Omega$ is the transversal intersection of bounded smooth pseudoconvex domains.
$\Omega=\Omega _1\setminus\overline\Omega _2$ where $\Omega _2$ is the union of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex convex domain with a piecewise smooth boundary.
$\Omega=\Omega _1\setminus\overline{\Omega}_2$ where $\Omega _2$ is the intersection of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex domain with a piecewise smooth boundary.
The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for $\overline{\partial}_b$ on domains with piecewise smooth boundaries in a pseudoconvex manifold.

References [Enhancements On Off] (What's this?)

  • 1. Barrett, D., Behavior of the Bergman projection on the Diederich-Fornaess worm, Acta Math. 168 (1992), 1-10. MR 93c:32033
  • 2. Boas, H. P., Straube, E. J., Sobolev estimates for the $\overline{\partial }$-Neumann operator on domains in $\mathbf{C}^{n}$ admitting a defining function that is plurisubharmonic on the boundary, Math. Zeit. 206 (1991), 81-88. MR 92b:32027
  • 3. Catlin, D., Subelliptic estimates for the $\overline \partial $-Neumann problem on pseudo-convex domains, Ann. of Math. 126 (1987), 131-191. MR 88i:32025
  • 4. Chaumat, J., Chollet, A.M., Noyaux pour résoudre l'équation $\overline{\partial }$ dans des classes ultra-
    différentiables sur des compacts irréguliers de $\mathbb{C}^{n}$
    , Princeton University Press (1993), 205-226. MR 94b:32018
  • 5. Christ, M., Global $C^{\infty }$ irregularity of the $\overline \partial $-Neumann problem for worm domains, Jour. Amer. Math. Soc. 9 (1986), 1171-1185. MR 96m:32014
  • 6. Diederich, K., Fornaess, J. E., Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), 129-141. MR 55:10728
  • 7. Dufresnoy, A., Sur L'opérateur d" et les fonctions différentiables au sens de Whitney, Ann. Inst. Fourier, Grenoble 29 (1979), 229-238. MR 80i:32050
  • 8. Grauert H., Lieb, I., Das Ramirezsche Integral und die Lösung der Gleichung $\overline \partial f=\alpha $ im Bereich der beschränkten Formen, Proc. Conf. Complex Analysis, Rice Univ. Studies 56 (1970), 29- 50. MR 42:7938
  • 9. Henkin, G.M., Integral representation of functions in strictly pseudoconvex domains and applications to the $\overline \partial $-problem, Math. USSR Sb. 7 (1969), 579-616. MR 40:2902
  • 10. Henkin, G.M., Uniform estimates for solutions to the $\overline \partial $-problem in Weil domains, Uspehi Mat. Nauk 26 (1971), 211-212 (Russian). MR 45:3753
  • 11. Henkin, G.M., The H. Lewy equation and analysis on pseudoconvex manifolds, Russian Math. Surveys 32 (1977), 59-130. MR 56:12318
  • 12. Henkin, G.M., Leiterer, J, Theory of functions on complex manifolds, Birkhäuser, Boston, Mass. (1984). MR 86a:32002
  • 13. Hörmander, L., $L^{2}$ estimates and existence theorems for the $\overline \partial $ operator, Acta Math. 113 (1965), 89-152. MR 31:3691
  • 14. Hortmann, M., Über die Lösbarkeit der $\overline \partial $-Gleichung mit Hilfe von $L^{p}$, $C^{k}$ und $D$'-stetigen Integraloperatoren, Math. Ann. 223 (1976), 139-156. MR 54:10674
  • 15. Kohn, J.J., Harmonic integrals on strongly pseudoconvex manifolds, I, Ann. of Math. 78 (1963), 112-148. MR 27:2999
  • 16. Kohn, J.J., Global regularity for $\overline{\partial }$ on weakly pseudoconvex manifolds, Trans. Am. Math. Soc. 181 (1973), 273-292. MR 49:9442
  • 17. J. J. Kohn, Subellipticity of the $\overline \partial $-Neumann problem on pseudoconvex domains: Sufficient conditions, Acta Math. 142 (1979), 79- 122. MR 80d:32020
  • 18. Lieb, I., Range, R.M., Lösungsoperatoren für den Cauchy-Riemann Komplex mit $C^{k}$-Abschätzungen, Math. Ann. 253 (1980), 145-164. MR 82:32012
  • 19. Ma, L., Michel, J., Local regularity for the tangential Cauchy-Riemann Complex, J. Reine Angew. Math 442 (1993), 63-90. MR 94h:32032
  • 20. Michel, J., Randregularität des $\overline{\partial }$-Problems für stückweise streng pseudokonvexe Gebiete in $\mathbb{C}^{n}$, Math. Ann 280 (1988), 46-68. MR 89f:32033
  • 21. Michel, J., Integral representations on weakly pseudoconvex domains, Math. Zeit. 208 (1991), 437-462. MR 93a:32005
  • 22. Michel, J., Perotti, A., $C^{k}$-regularity for the $\overline{\partial }$-equation on strictly pseudoconvex domains with piecewise smooth boundaries, Math. Zeit. 203 (1990), 415-427. MR 91b:32019
  • 23. Michel, J., Perotti, A., $C^{k}$-regularity for the $\overline{\partial }$-equation on a piecewise smooth union of strictly pseudoconvex domains in $\mathbb{C}^{n}$, Ann. Sc. Norm. Sup. Pisa 21 (1994), 483-495. MR 95m:32026
  • 24. Michel, J., Shaw, M.-C., Subelliptic estimates for the $\overline \partial $-Neumann operator on piecewise smooth strictly pseudoconvex domains, Duke Math. Jour. 93 (1998), 115-129. MR 99b:32019
  • 25. Michel, J., Shaw, M.-C., A decomposition problem on weakly pseudoconvex domains, Math. Zeit. 230 (1999), 1-19.
  • 26. Michel, J., Shaw, M.-C., $C^{\infty }$-regularity of solutions of the tangential CR-equations on weakly pseudoconvex manifolds, Math. Ann. 311 (1998), 147-162. CMP 98:13
  • 27. Polyakov, P.L., On Banach cohomologies of stratified spaces, Uspehi Mat. Nauk 26 (1971), 243-244 (Russian). MR 45:3781
  • 28. Polyakov, P.L., Banach cohomology of piecewise strictly pseudoconvex domains, Math. USSR Sb. 17 (1972), 237-256. MR 46:396
  • 29. Range, R.M., Holomorphic functions and integral representations in several complex variables, Graduate Texts in Math, Springer-Verlag 108 (1986). MR 87i:32001
  • 30. Range, R. M., Siu, Y. T., Uniform estimates for the $\overline{\partial }$-equation on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann. 83 (1973), 325-354. MR 49:3214
  • 31. Shaw, M.-C., Global solvability and regularity for $\overline{\partial }$ on an annulus between two weakly pseudoconvex domains, Trans. Amer. Math. Soc. 291 (1985), 255-267. MR 86m:32030
  • 32. Shaw, M.-C., $L^{p}$ estimates for local solutions of $\overline \partial _{b}$ on strongly pseudoconvex CR manifolds, Math. Ann. 288 (1990), 35-62. MR 92b:32028
  • 33. Shaw, M.-C., Local existence theorems with estimates for $\overline{\partial }_{b} $ on weakly pseudoconvex boundaries, Math. Ann. 294 (1992), 677-700. MR 94b:32026
  • 34. Shaw, M.-C., Semi-global existence theorems of $ \overline{\partial }_{b} $ for $(0,n-2)$ forms on pseudoconvex boundaries in $ \mathbb{C}^{n} $, Colloque D'Analyse Complexe et Géométrie, Astérisque, No. 217 (1993), 227-240. MR 95a:32028
  • 35. Stein E. M., Singular integrals and differentiability properties of functions, Princeton University Press. Princeton, New Jersey (1970). MR 44:7280
  • 36. Vassiliadou, S., Homotopy fomulas for $\overline \partial _{b}$ and subelliptic estimates for the $\overline \partial $-Neumann problem, Thesis, Notre Dame (1997).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35N05, 35N10, 32F10

Retrieve articles in all journals with MSC (1991): 35N05, 35N10, 32F10

Additional Information

Joachim Michel
Affiliation: Université du Littoral, Centre Universitaire de la Mi-Voix, F-62228 Calais, France

Mei-Chi Shaw
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Keywords: Cauchy-Riemann equations, piecewise smooth boundary, tangential Cauchy-Riemann equations.
Received by editor(s): August 11, 1997
Received by editor(s) in revised form: May 7, 1998
Published electronically: July 9, 1999
Additional Notes: Partially supported by NSF grant DMS 98-01091
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society