The Pullback Equation for Differential Forms

The Pullback Equation for Differential Forms

Volumn , Issue , 2012, Pages 1-436

  Csató, Gyula ^a   Dacorogna, Bernard ^a   Kneuss, Olivier ^a  

^a NONE   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85020718709     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-0-8176-8313-9     Document Type: Book

References (135)

1. Abraham R., Marsden J.E. and Ratiu T., Manifolds, tensor analysis, and applications, second edition, Springer-Verlag, New York, 1988.
2. Adams R.A., Sobolev spaces, Academic Press, New York, 1975; second edition with Fournier J.J.F., 2003.
3. Agmon S., Douglis A. and Nirenberg L., Estimates near the boundary for solutions of elliptic partial differentialequations satisfying general boundary conditions I, Commun. Pure Appl. Math., 12 (1959), 623-727.
4. Agmon S., Douglis A. and Nirenberg L., Estimates near the boundary for solutions of elliptic partial differentialequations satisfying general boundary conditions II, Commun. Pure Appl. Math., 17 (1964), 35-92.
5. Alt H.W., Lineare Funktionalanalysis, fourth edition, Springer-Verlag, Berlin, 2002.
6. Amrouche C., Bernardi C., Dauge M., Girault V., Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.
7. Bandyopadhyay S., private communication, 2010.
8. Bandyopadhyay S. and Dacorogna B., On the pullback equation φ* (g) = f , Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1717-1741.
9. Bandyopadhyay S., Dacorogna B. and Kneuss O., The pullback equation for degenerate forms, Discrete Continuous Dynami. Syst. A, 27 (2010), 657-691.
10. Banyaga A., Formes-volume sur les variétés à bord, Enseignement Math., 20 (1974), 127-131.
11. Barbarosie C., Representation of divergence-free vector fields, Q. Appl. Math., to appear.
12. Bogovski M.E., Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl., 20 (1979), 1094-1098.
13. Bolik J., H Weyl's boundary value problems for differential forms, Differ. Integ. Eqs., 14 (2001), 937-952.
14. Borchers W. and Sohr H., On theequations rot v = g and divu = f with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87.
15. Bourbaki N., Éléments de mathématique. Algèbre. Chapitres 1 à 3, Hermann, Paris, 1970.
16. Bourgain J. and Brézis H., Sur l'équation divu = f , C. R. Acad. Sci. Paris Sér. I Math., 334 (2002), 973-976.
17. Brézis H., Analyse fonctionelle, théorie et applications, Dunod, Paris, 1999.
18. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L. and Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications 18, Springer-Verlag, New York, 1991.
19. Burago D. and Kleiner B., Separated nets in Euclidean space and Jacobian of biLipschitz maps, Geom. Funct. Anal., 8 (1998), 273-282.
20. Calderon A.P., Lebesgue spaces of differentiable functions and distributions, Proc. Symp. Pure Math., Vol. IV, American Mathematical Society, Providence, RI, 1961, pp. 33-49.
21. Cartan E., Selecta, Gauthier-Villars, Paris, 1939.
22. Coddington E.A. and Levinson N., Theory of ordinary differentialequations, McGraw-Hill Book Company Inc., New York, 1955.
23. Csató G., PhD thesis, EPFL Lausanne (2012).
24. Csató G. and Dacorogna B., An identity involving exterior derivatives and applications to Gaffney inequality, Discrete Continuous Dynam. Syst., to appear.
25. Cupini G., Dacorogna B. and Kneuss O., On the equation det▿u = f with no sign hypothesis, Calc. Var. Partial Differ. Equ., 36 (2009), 251-283.
26. Dacorogna B., A relaxation theorem and its applications to the equilibrium of gases, Arch. Rational Mech. Anal., 77 (1981), 359-386.
27. Dacorogna B., Existence and regularity of solutions of dw = f with Dirichlet boundary conditions, Nonlinear Problems in Mathematical Physics and Related Topics, Int. Math. Ser. (N. Y.), 1, Kluwer/Plenum, New York (2002), 67-82.
28. Dacorogna B., Direct methods in the calculus of variations, second edition, Springer-Verlag, New York, 2007.
29. Dacorogna B., Introduction to the calculus of variations, second edition, Imperial College Press, London, 2009.
30. 0, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 14 (2003), 239-245.
31. Dacorogna B. and Kneuss O., Divisibility in Grassmann algebra, Linear Multilinear Alg., to appear.
32. Dacorogna B. and Kneuss O., A global version of Darboux theorem with optimal regularity and Dirichlet condition, Advan. Differ. Eq., 16 (2011), 325-360.
33. Dacorogna B. and Moser J., On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Lineaire, 7 (1990), 1-26.
34. Darboux G., Sur le problème de Pfaff, Bull Sci.Math. 6 (1882), 14-36, 49-68.
35. Dautray R. and Lions J.L., Analyse mathématique et calcul numerique, Masson, Paris, 1988.
36. de la Llave R. and Obaya R., Regularity of the composition operator in spaces of Hölder functions, Discrete Continuous Dynami. Syst. A, 5 (1999), 157-184.
37. do Carmo M. P., Differential forms and applications, Universitext, Springer-Verlag, Berlin, 1994.
38. Duff G.F. and Spencer D.C., Harmonic tensors on Riemannian manifolds with boundary, Ann. Math., 56 (1952), 128-156.
39. Duvaut G. and Lions J.L., Inequalities in mechanics and physics, Springer-Verlag, Berlin, 1976.
40. Edmunds D.E. and Evans W.D., Spectral theory and differential operators, Oxford Science Publications, Oxford, 1987.
41. Evans L.C., Partial differentialequations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
42. Fefferman C., Whitney's extension problems and interpolation of data, Bull. Amer. Math. Soc., 46 (2009), 207-220.
43. Fonseca I. and Gangbo W., Degree theory in analysis and applications, Oxford University Press, New York, 1995.
44. Gaffney M.P., The harmonic operator for exterior differential forms, Proc. Natl. Acad. Sci. U. S. A., 37 (1951), 48-50.
45. Gaffney M.P., Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc, 78 (1955), 426-444.
46. Galdi G.P., An introduction to the mathematical theory of the Navier-Stokesequations, Springer-Verlag, New York, 1994.
47. Georgescu V., Some boundary value problems for differential forms on compact Riemannian manifolds, Ann. Mat. Pura Appl. 122 (1979), 159-198.
48. Gilbarg D. and Hormander L., Intermediate Schauder estimates, Arch. Rational Mech. Anal., 74 (1980), 297-318.
49. Gilbarg D. and Trudinger N.S., Elliptic partial differentialequations of second order, Springer-Verlag, Berlin, 1977.
50. Girault V. and Raviart P.A., Finite element approximation of the Navier-Stokesequations, Lecture Notes in Math. 749, Springer-Verlag, Berlin, 1979.
51. Godbillon C., Géométrie différentielle et mécanique analytique, Hermann, Paris, 1969.
52. Godement R., Cours d'algèbre, Hermann, Paris, 1963.
53. Golubitsky M. and Tischler D., A survey on the singularites and stability of differential forms, Astérisque, 59-60, Société Mathématique de France, Paris, 1978.
54. Greub W. H., Mutilinear algebra, Springer-Verlag, New York, 1967.
55. Hormander L., The boundary problems of physical geodesy, Arch. Rational Mech. Anal., 62 (1976), 1-52.
56. Horn R.A. and Johnson C.A., Matrix analysis, Cambridge University Press, Cambridge, 1985.
57. Iwaniec T. and Martin G., Geometric function theory and non-linear analysis, Oxford University Press, Oxford, 2001.
58. Iwaniec T., Scott C. and Stroffolini B., Nonlinear Hodge theory on manifolds with boundary, Annali Mat. Pura Appl., 177 (1999), 37-115.
59. Kapitanskii L.V. and Pileckas K., Certain problems of vector analysis, J. Soviet Math., 32 (1986), 469-483.
60. Kneuss O., PhD thesis, EPFL Lausanne (2011).
61. Krantz S.G. and Parks H.R., The geometry of domains in space, Birkhauser, Boston, 1999.
62. Krantz S.G. and Parks H.R., The implicit function theorem. History, theory and applications, Birkhauser, Boston, 2002.
63. Kress R., Potentialtheoretische Randwertprobleme bei Tensorfelderrn beliebiger Dimensions und beliebigen Ranges, Arch. Rational Mech. Anal., 47 (1972), 59-80.
64. Ladyzhenskaya O.A., The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1969.
65. Ladyzhenskaya O.A. and Solonnikov V.A., Some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokesequations, J. Soviet Math., 10 (1978), 257-286.
66. Ladyzhenskaya O.A. and Uraltseva N.N., Linear and quasilinear ellipticequations, Academic Press, New York, 1968.
67. Lang S., Algebra, Springer-Verlag, New York, 2002.
68. Lee J.M., Introduction to smooth manifolds, Springer-Verlag, New York, 2003.
69. Lieberman G.M., The quasilinear Dirichlet problem with decreased regularity at the boundary, Commun. Partial Differ. Equ., 6 (1981), 437-497.
70. Lieberman G.M., The Dirichlet problem for quasilinear ellipticequations with Holder continuous boundary values, Arch. Rational Mech. Anal., 79 (1982), 305-323.
71. Martinet J., Sur les singularités des formes différentielles, Ann. Inst. Fourier, 20 (1970), 95-178.
72. Mc Duff D. and Salamon D., Introduction to symplectic topology, second edition, Oxford Science Publications, Oxford, 1998.
73. Mc Mullen C.T., Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal., 8 (1998), 304-314.
74. Marcus M., Finite dimensional multilinear algebra. Part II, Marcel Dekker, New York, 1975.
75. Meisters G.H. and Olech C., Locally one-to-one mappings and a classical theorem on schlicht functions, Duke Math. J., 30 (1970), 63-80.
76. Morrey C.B., A variational method in the theory of harmonic integrals II, Amer. J. Math., 78 (1956), 137-170.
77. Morrey C.B., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966.
78. Moser J., On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.
79. Necas J., Les méthodes directes en theorie des équations elliptiques, Masson, Paris, 1967.
80. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
81. Pelletier F., Singularités d'ordre supérieur de 1-formes, 2-formes et équations de Pfaff, Inst. Hautes Etudes Sci. Publ. Math., 61 (1985), 129-169.
82. Postnikov M., Leçons de géométrie: algèbre linéaire et géométrie différentielle, Mir, Moscow, 1988.
83. Preiss D., Additional regularity for Lipschitz solutions of pde, J. Reine Angew. Math., 485 (1997), 197-207.
84. Reimann H.M., Harmonische Funktionen und Jacobi-Determinanten von Diffeomorphismen, Comment. Math. Helv., 47 (1972), 397-408.
85. Rivière T. and Ye D., Resolutions of the prescribed volume form equation, Nonlinear Differ. Eq. Appl., 3 (1996), 323-369.
86. Roussarie R., Modèles locaux de champs et de formes, Astérisque, 30, Societe Mathematique de France, Paris, 1975.
87. Rudin W., Real and complex analysis, Mc Graw-Hill, New York, 1966.
88. Schwartz J.T., Nonlinear functional analysis, Gordon and Breach, New York, 1969.
89. Schwarz G., Hodge decomposition-A method for solving boundary value problems, Lecture Notes in Math. 1607, Springer-Verlag, Berlin, 1995.
90. Serre, D., Matrices. Theory and applications, Springer-Verlag, New York, 2002.
91. Spivak M., A comprehensive introduction to differential geometry, third edition, Publish or Perish, Houston, TX, 1999.
92. Stein E.M., Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, 1970.
93. Sternberg S., Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, NJ, 1964.
94. Tartar L., Topics in nonlinear analysis, Preprint, University of Wisconsin, Madison, 1975.
95. Tartar L., unpublished, 1978.
96. Taylor M.E., Partial differentialequations, Vol. 1, Springer-Verlag, New York, 1996.
97. Turiel F.J., Some remarks concerning the transitive geometric structures defined by exterior forms, Seminar on Differential Geometry, 1981-1982, Exp. No. 7, Université des Sciences et Techniques du Languedoc, Montpellier, 1982.
98. Turiel F.J., Classification locale des 3-formes fermees infinitesimalement transitives à cinq variables, Cahiers Mathematiques, 30, Université des Sciences et Techniques du Languedoc, Montpellier, 1984.
99. Turiel F.J., Un théorème de Darboux pour un couple de formes symplectiques. Feuilletages riemanniens, quantification géométrique et mecanique (Lyon, 1986), Hermann, Paris, 1988. pp. 103-121.
100. Turiel F.J., Classification locale simultanée de deux formes symplectiques compatibles, Manuscr. Math. 82 (1994), 349-362.
101. 5, Feuilletages et systèmes intégrables (Montpellier, 1995), Progr. Math., 145, Birkhäuser Boston, 1997, pp. 157-168.
102. Turiel F.J., Sur certaines n-formes en dimension 2n, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 471-474.
103. Von Wahl W., On necessary and sufficient conditions for the solvability of theequations rotu = γ and divu = ε with u vanishing on the boundary, Lecture Notes in Math. 1431, Springer-Verlag, Berlin, 1990, pp. 152-157.
104. Von Wahl W., Vorlesung über das Aussenraumproblem für die instationären Gleichungen von Navier-Stokes; Rudolph-Lipschitz-Vorlesung. Sonderforschungsbereich 256. Nichtlineare Partielle Differentialgleichungen, Bonn, 1989.
105. Widman K.O., Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differentialequations, Math. Scand., 21 (1967), 17-37.
106. Ye D., Prescribing the Jacobian determinant in Sobolev spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 275-296.
107. Zehnder E., Note on smoothing symplectic and volume preserving diffeomorphisms, Lecture Notes in Mathematics 597, Springer-Verlag, Berlin, 1976, pp. 828-855.
108. Zhitomirskîi M.Y., Degeneration of differential 1-forms and Pfaffian structures, Usp. Mat. Nauk 46 (1991), 47-78; translation in Russian Math. Surveys 46 (1991), 53-90.
109. Zhitomirskîi M.Y., translation in Russian Math. Surveys 46 (1991), 53-90
110. Abraham R. and Marsden J. E., Foundations of mechanics, second edition, Advanced Book Program, Benjamin/Cummings Publishing Co., Inc., Reading, MA, 1978.
111. Bachman D., A Geometric approach to differential forms, Birkhäuser, Boston, 2006.
112. Bolik J., Boundary value problems for differential forms on compact Riemannian manifolds, Part II, Analysis (Munich) 27 (2007), 477-493.
113. Bourgain J. and Brézis H., New estimates for ellipticequations and Hodge type systems, J. Eur. Math. Soc., 9 (2007), 277-315.
114. Brézis H. and Nguyen H.M., The Jacobian determinant revisited, preprint (2010).
115. n in fractional Sobolev and Hölder spaces, preprint (2010).
116. Cartan E., Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, Paris, 1945.
117. Choquet-Bruhat Y., De Witt-Morette C. and Dillard-Bleick M., Analysis, manifolds and physics, North-Holland, Amsterdam, 1977.
118. Debever R., Quelques problèmes d'équivalence de formes différentielles alternées, Acad. Roy. Belgique Bull. Cl. Sci., 31 (1946), 262-277.
119. Deheuvels R., Tenseurs et spineurs, Presses Universitaires de France, Paris, 1993.
120. De Rham G., Differentiable manifolds, Springer-Verlag, Berlin, 1984.
121. Flanders H., Differential forms with applications to the physical sciences, Dover, New York, 1989.
122. Folland G.B., Introduction to partial differentialequations, Princeton University Press, Princeton, NJ, 1976.
123. 3. Differential geometry, global analysis, and topology (Halifax, NS, 1990), CMS Conf. Proc. Amer. Math. Soc. 12, American Mathematical Society, Providence, RI, 1991, pp. 41-50.
124. Giaquinta M., Modica G. and Soucek J., Cartesian currents in the calculus of variations 1, Springer-Verlag, Berlin, 1998.
125. Hörmander L., Linear partial differential operators, Springer-Verlag, Berlin, 1963.
126. Ivey T.A. and Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, 61, American Mathematical Society, Providence, RI, 2003.
127. John F., Partial differentialequations, Springer-Verlag, Berlin, 1982.
128. Kobayashi S. and Nomizu K., Foundations of differential geometry. Vol. I, John Wiley & Sons Inc., New York, 1996.
129. Lions J.L. and Magenes E., Non-homogeneous boundary value problems and applications I,II,III, Springer-Verlag, Berlin, 1972.
130. Mitrea D. and Mitrea M., Boundary integral methods for harmonic differential forms in Lipschitz domains, Electron. Res. Announce. Ameri. Math. Soc., 2 (1996), 92-922. Morrey C.B. and Eells J., A variational method in the theory of harmonic integrals, Ann. Math., 63 (1956), 91-128.
131. Morrey C.B. and Eells J., A variational method in the theory of harmonic integrals, Ann. Math., 63 (1956), 91-128
132. Papy G., Sur l'arithmétique dans les algèbres de Grassman, Acad. Roy. Belgique. Cl. Sci. Mem. Coll., 26 (1952), 1-108.
133. Picard R., An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Zeitschr., 187 (1984), 151-164.
134. Triebel H., Theory of function spaces II, Monographs in Mathematics 84, Birkhäuser, Basel, 1992.
135. Von Wahl W., Estimating ▿u by divu and curlu, Math. Methods Appl. Sci., 15 (1992), 123-143.