The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains

Communications on Pure and Applied Analysis

Volumn 7, Issue 6, 2008, Pages 1295-1333

  Mitrea, Dorina ^a   Mitrea, Marius ^a   Monniaux, Sylvie ^b  

^a UNIVERSITY OF MISSOURI   (United States)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

Besov; Differential forms; Dirichlet condition; Divergence equation; Exterior derivative; Lipschitz domain; Poisson problem; Sobolev; Triebel Lizorkin spaces

Indexed keywords

EID: 57149120594     PISSN: 15340392     EISSN: 15535258     Source Type: Journal    
DOI: 10.3934/cpaa.2008.7.1295     Document Type: Article

References (61)

1. p-integrability of the second order derivatives of Green potentials in convex domains, Pacific J. Math., 159 (1993), 201-225.
2. D. N. Arnold, L. R. Scott and M. Vogelius, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15 (1988), 169-192.
3. M. S. Birman and M. Z. Solomyak, The Maxwell operator in domains with nonsmooth boundary, Sibirsk. Mat. Zh., 28 (1987), 23-36.
4. M. S. Birman and M. Z. Solomyak, Weyl asymptotics of the spectrum, of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya, 20 (1987), 23-28.
5. M. S. Birman and M. Z. Solomyak, Principal singularities of the electric component of the electromagnetic field in regions with screens, Algebra and Analysis (St. Petersburg Mathematical Journal), 5 (1993), 143-159.
6. M. E. Bogovskiǐ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR, 248 (1979), 1037-1040.
7. M. E. Bogovskiǐ, Solution of some vector analysis problems connected with operators div and grad, Trudy Seminar N. L. Sobolev, Akademia Nauk SSSR, Sibirskoe Otdelenie Matematiki, Nowosibirsk, 80 (1980), 5-40.
8. 2 decay for the Navier-Stokes flows in exterior domains, Acta Math., 165 (1990), 189-227.
9. W. Borchers and H. Sohr, On the equations rot u = g and div u = f with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87.
10. J. Bourgain and H. Brezis, Sur l'équation div u = f, C. R. Math. Acad. Sci. Paris, 334 (2002), 973-976.
11. J. Bourgain and H. Brezis, On the equation div Y = f and application to control of phases, J. Amer. Math. Soc, 16 (2003), 393-426.
12. A. Caetano, Approximation by functions of compact support in Besov-Triebel-Lizorkin spaces on irregular domains, Studia Math., 142 (2000), 47-63.
13. B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 1-26.
14. B. Dacorogna, Existence and regularity of solutions of dw = f with Dirichlet boundary conditions, in "Nonlinear Problems in Mathematical Physics and Related Topics, I," Kluwer/Plenum, New York, (2002) 67-82.
15. 0, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 14 (2003), 239-245.
16. R. Dautray and J.-L. Lions, "Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques," 5, Masson, Paris, 1988.
17. p(.) and problems related to fluid dynamics, J. Reine Angew. Math., 563 (2003), 197-220.
18. R. G. Durán and M. A. Muschietti, An explicit right inverse of the divergence operator which is continuous in weighted norms, Studia Math., 148 (2001), 207-219.
19. G. Duvaut and J.-L. Lions, "Les Inéquations en Mécanique et en Physique," Travaux et Recherches Mathematiques, 21, Dunod, Paris, 1972.
20. N. D. Filonov, On bounded solutions to the equation divu = f in a plane domain, in "Proceedings of the St. Petersburg Mathematical Society," VI, Amer. Math. Soc. Transl. Ser. 2, 199, Amer. Math. Soc, Providence, RI, (2000), 199-207.
21. S. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc, 119 (1993), 225-233.
22. G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations," Springer-Verlag, New York, 1994.
23. V. Girault and P.-A. Raviart, "Finite Element Approximation of the Navier-Stokes Equations," Lecture Notes in Mathematics, 749, Springer-Verlag, Berlin-New York, 1979.
24. Ft. Godement, "Topologie Algébrique et Théorie des Faisceaus," Paris, Hermann, 1958.
25. q with respect to the operator rot, Math. Ann., 288 (1990), 245-262.
26. T. Iwaniec and A. Lutoborski, Integral estimates for null Lagrangians, Arch. Rational Mech. Anal., 125 (1993), 25-79.
27. D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.
28. n," Mathematical Reports, Harwood Academic, New York, 1984.
29. G. A. Kalyabin, Theorems on extension, multipliers and diffeomorphisms for generalized Sobolev-Liouville classes in domains with Lipschitz boundary, Trudy Mat. Inst. Steklov., 172 (1985), 173-186.
30. L. V. Kapitanskiǐand K. I. Piletskas, Certain problems of vector analysis, J. Soviet Math., 32 (1986), 469-483.
31. T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1976.
32. O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Gordon and Breach, Science Publishers, New York-London-Paris, 1969.
33. O. A. Ladyzhenskaya and V. A. Solonnikov, Some problems of vector analysis, and generalized formulations of boundary value problems for the Navier-Stokes equation, Journal of Soviet Mathematics, 10 (1978), 257-286.
34. N, Trans. Amer. Math. Soc, 357 (2005), 1469-1496.
35. E. Magenes and G. Stampacchia, I problemi al contorno per le equazioni differenziali di tipo ellittico, Ann. Scuola Norm. Sup. Pisa (3), 12 (1958), 247-358.
36. W. S. Massey, "Singular Homology Theory," Springer-Verlag, 1980.
37. S. Mayboroda and M. Mitrea, The solution of the Chang-Krein-Stein conjecture, in "Proceedings of the Conference in Harmonic Analysis and its Applications," (A. Miyachi ed.), Tokyo Woman's Cristian University, Tokyo, Japan, (2007), 61-154.
38. V. G. Maz'ya, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
39. V. Maz'ya, M. Mitrea and T. Shaposhnikova, The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients, arxiv:math/0701898
40. D. Mitrea, M. Mitrea and M. Taylor, "Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds," Mem. Amer. Math. Soc, 150 no. 713, 2001.
41. D. Mitrea and M. Mitrea, Finite energy solutions of Maxwell's equations and constructive Hodge decompositions on nonsmooth Riemannian manifolds, J. Funct. Anal., 190 (2002), 339-417.
42. M. Mitrea, Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds, Duke Math. J., 125 (2004), 467-547.
43. M. Mitrea and S. Monniaux, The regularity of the Stokes operator and the Pujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains, J. Funct. Anal., 254 (2008), 1522-1574.
44. M. Mitrea, M. Taylor and A. Vasy, Lipschitz domains, domains with corners, and the Hodge Laplacian, Comm. Partial Differential Equations, 30 (2005), 1445-1462.
45. J. Nečas, "Les Méthodes Directes en Théorie des Équations Elliptiques," Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967.
46. W. Rudin, "Functional Analysis," 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
47. T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators," de Gruyter, Berlin, New York, 1996.
48. V. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc. (2), 60 (1999), 237-257.
49. G. Schwarz, "Hodge Decomposition-A Method for Solving Boundary Value Problems," Lecture Notes in Mathematics, 1607 Springer-Verlag, 1995.
50. G. Schwarz, The existence of solutions of a general boundary value problem for the divergence, Math. Methods Appl. Sci., 17 (1994), 95-105.
51. M. Spivak, "Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus," W. A. Benjamin, Inc., New York-Amsterdam, 1965.
52. E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, 30 Princeton University Press, Princeton, N.J., 1970.
53. L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay, 1978.
54. M. E. Taylor, "Partial Differential Equations," I-III Springer-Verlag, 1996.
55. R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis" (revised edition), Studies in Mathematics and its Applications, 2 North-Holland Publishing Co., Amsterdam-New York, 1979.
56. H. Triebel, Theory of Function Spaces II, Monographs in Mathematics, 84 Birkhäuser Verlag, Basel, 1992.
57. H. Triebel, "The Structure of Functions," Monographs in Mathematics, 97 Birkhäuser Verlag, Basel, 2001.
58. H. Triebel, Function spaces on Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers, Rev. Mat. Complut., 15 (2002), 475-524.
59. W. von Wahl, On necessary and sufficient conditions for the solvability of the equations rot u = γ and div u = ∈ with u vanishing on the boundary, in "Lecture Notes in Math.," Springer, Berlin, 1431 (1990), 152-157.
60. X. M. Wang, A remark on the characterization of the gradient of a distribution, Appl. Anal., 51 (1993), 35-40.
61. F. Warner, "Foundations of Differentiable Manifolds and Lie Groups," Graduate Texts in Mathematics, 94 Springer-Verlag, 1983.